{"id":32862,"date":"2022-04-27T13:00:44","date_gmt":"2022-04-27T13:00:44","guid":{"rendered":"http:\/\/toposuranos.com\/material\/?p=32862"},"modified":"2025-04-03T23:10:12","modified_gmt":"2025-04-03T23:10:12","slug":"%e0%a4%b8%e0%a4%be%e0%a4%a7%e0%a4%be%e0%a4%b0%e0%a4%a3-%e0%a4%85%e0%a4%b5%e0%a4%95%e0%a4%b2-%e0%a4%b8%e0%a4%ae%e0%a5%80%e0%a4%95%e0%a4%b0%e0%a4%a3-edo-%e0%a4%95%e0%a5%8d%e0%a4%af%e0%a4%be-%e0%a4%b9","status":"publish","type":"post","link":"http:\/\/toposuranos.com\/material\/hi\/%e0%a4%b8%e0%a4%be%e0%a4%a7%e0%a4%be%e0%a4%b0%e0%a4%a3-%e0%a4%85%e0%a4%b5%e0%a4%95%e0%a4%b2-%e0%a4%b8%e0%a4%ae%e0%a5%80%e0%a4%95%e0%a4%b0%e0%a4%a3-edo-%e0%a4%95%e0%a5%8d%e0%a4%af%e0%a4%be-%e0%a4%b9\/","title":{"rendered":"\u0938\u093e\u0927\u093e\u0930\u0923 \u0905\u0935\u0915\u0932 \u0938\u092e\u0940\u0915\u0930\u0923 (EDO) \u0915\u094d\u092f\u093e \u0939\u0948?"},"content":{"rendered":"<style>\np, ul, ol{\ntext-align: justify;\n}\nh1{\ntext-align:center;\ntext-transform: uppercase;\n}\nh2{\ntext-align:center;\ntext-transform: uppercase;\nfont-size:24pt;\n}\nh3 { \n    text-align: center;\n    text-transform: uppercase;\n    font-size: 24px !important;\n}\n<\/style>\n<h1>\u0938\u093e\u0927\u093e\u0930\u0923 \u0905\u0935\u0915\u0932 \u0938\u092e\u0940\u0915\u0930\u0923 (EDO) \u0915\u094d\u092f\u093e \u0939\u0948?<\/h1>\n<p style=\"text-align:center;\" dir=\"ltr\"><em><strong>\u0938\u093e\u0930\u093e\u0902\u0936:<\/strong><\/br>\u0907\u0938 \u0915\u0915\u094d\u0937\u093e \u092e\u0947\u0902, \u0915\u094d\u0930\u092e k \u0915\u0947 \u0938\u093e\u0927\u093e\u0930\u0923 \u0905\u0935\u0915\u0932 \u0938\u092e\u0940\u0915\u0930\u0923\u094b\u0902 (EDO) \u0915\u093e \u0905\u0928\u094d\u0935\u0947\u0937\u0923 \u0915\u093f\u092f\u093e \u0917\u092f\u093e \u0939\u0948, \u0907\u0938\u0915\u0940 \u092a\u0930\u093f\u092d\u093e\u0937\u093e \u0914\u0930 \u0938\u093e\u092e\u093e\u0928\u094d\u092f \u090f\u0935\u0902 \u092e\u093e\u0928\u0915 \u0930\u0942\u092a \u092e\u0947\u0902 \u0909\u0938\u0915\u0940 \u0905\u092d\u093f\u0935\u094d\u092f\u0915\u094d\u0924\u093f \u0938\u0947 \u0936\u0941\u0930\u0941\u0906\u0924 \u0915\u0930\u0924\u0947 \u0939\u0941\u090f\u0964 \u091c\u0948\u0915\u094b\u092c\u093f\u092f\u0928 \u092e\u0948\u091f\u094d\u0930\u093f\u0915\u094d\u0938 \u0914\u0930 \u0905\u0902\u0924\u0930\u094d\u0928\u093f\u0939\u093f\u0924 \u092b\u0932\u0928 \u092a\u094d\u0930\u092e\u0947\u092f \u091c\u0948\u0938\u0947 \u0905\u0935\u0927\u093e\u0930\u0923\u093e\u0913\u0902 \u0915\u0947 \u092e\u093e\u0927\u094d\u092f\u092e \u0938\u0947, \u0907\u0928 \u0938\u092e\u0940\u0915\u0930\u0923\u094b\u0902 \u0915\u0947 \u0939\u0932 \u0914\u0930 \u0938\u0902\u092c\u0926\u094d\u0927 \u0917\u0941\u0923\u094b\u0902 \u0915\u094b \u0938\u092e\u091d\u0928\u0947 \u0915\u0940 \u0928\u0940\u0902\u0935 \u0930\u0916\u0940 \u091c\u093e\u0924\u0940 \u0939\u0948, \u091c\u0948\u0938\u0947 \u092a\u0930\u093f\u092d\u093e\u0937\u093e \u0915\u093e \u0921\u094b\u092e\u0947\u0928 \u0914\u0930 \u0938\u094d\u092a\u0937\u094d\u091f \u0935 \u0905\u092a\u094d\u0930\u0924\u094d\u092f\u0915\u094d\u0937 \u0939\u0932\u0964<\/em><\/p>\n<p style=\"text-align:center\"><strong>\u0905\u0927\u094d\u092f\u092f\u0928 \u0915\u0947 \u0909\u0926\u094d\u0926\u0947\u0936\u094d\u092f<\/strong><\/p>\n<p>\u0907\u0938 \u0915\u0915\u094d\u0937\u093e \u0915\u0947 \u0905\u0902\u0924 \u092e\u0947\u0902 \u091b\u093e\u0924\u094d\u0930 \u0938\u0915\u094d\u0937\u092e \u0939\u094b\u0917\u093e:<\/p>\n<ol>\n<li><strong>\u0938\u094d\u092e\u0930\u0923 \u0915\u0930\u0928\u093e<\/strong> \u090f\u0915 \u0938\u093e\u0927\u093e\u0930\u0923 \u0905\u0935\u0915\u0932 \u0938\u092e\u0940\u0915\u0930\u0923 (EDO) \u0915\u0940 \u092a\u0930\u093f\u092d\u093e\u0937\u093e \u0914\u0930 \u0907\u0938\u0915\u0940 \u092e\u0942\u0932\u092d\u0942\u0924 \u0935\u093f\u0936\u0947\u0937\u0924\u093e\u0913\u0902 \u0915\u094b\u0964<\/li>\n<li><strong>\u0935\u094d\u092f\u093e\u0916\u094d\u092f\u093e \u0915\u0930\u0928\u093e<\/strong> \u090f\u0915 EDO \u0914\u0930 \u0909\u0938\u0915\u0947 \u0938\u0902\u092d\u093e\u0935\u093f\u0924 \u0939\u0932\u094b\u0902 \u0915\u0947 \u092c\u0940\u091a \u0915\u0947 \u0938\u0902\u092c\u0902\u0927 \u0915\u094b\u0964<\/li>\n<\/ol>\n<p style=\"text-align:center;\" dir=\"ltr\"><strong>\u0905\u0928\u0941\u0915\u094d\u0930\u092e\u0923\u093f\u0915\u093e<\/strong><br \/>\n<a href=\"#LaEcuacionDiferencialOrdinariaDeOrdenK\"><strong>\u0915\u094d\u0930\u092e k \u0915\u093e \u0938\u093e\u0927\u093e\u0930\u0923 \u0905\u0935\u0915\u0932 \u0938\u092e\u0940\u0915\u0930\u0923 (EDO)<\/strong><\/a><br \/>\n<a href=\"#TeoremaDeLaFuncionImplicita\">\u0905\u0902\u0924\u0930\u094d\u0928\u093f\u0939\u093f\u0924 \u092b\u0932\u0928 \u0915\u093e \u092a\u094d\u0930\u092e\u0947\u092f<\/a><br \/>\n<a href=\"#LaSolucionDeUnaEcuacionDiferencialOrdinaria\"><strong>\u0938\u093e\u0927\u093e\u0930\u0923 \u0905\u0935\u0915\u0932 \u0938\u092e\u0940\u0915\u0930\u0923 \u0915\u093e \u0939\u0932<\/strong><\/a><br \/>\n<a href=\"#CuidadoConElDominioDeDefinicionDeLasSoluciones\">\u0939\u0932 \u0915\u0947 \u092a\u0930\u093f\u092d\u093e\u0937\u093e \u0915\u094d\u0937\u0947\u0924\u094d\u0930 \u0915\u0947 \u092a\u094d\u0930\u0924\u093f \u0938\u093e\u0935\u0927\u093e\u0928\u0940<\/a><br \/>\n<a href=\"#SolucionExtendidaYSolucionMaximal\">\u0935\u093f\u0938\u094d\u0924\u093e\u0930\u093f\u0924 \u0939\u0932 \u0914\u0930 \u0905\u0927\u093f\u0915\u0924\u092e \u0939\u0932<\/a><br \/>\n<a href=\"#SolucionExplicitaYSolucionImplicita\">\u0938\u094d\u092a\u0937\u094d\u091f \u0939\u0932 \u0914\u0930 \u0905\u092a\u094d\u0930\u0924\u094d\u092f\u0915\u094d\u0937 \u0939\u0932<\/a>\n<\/p>\n<p><center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/zE29azRIKng\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen><\/iframe><\/center><\/p>\n<p>\u0905\u092c \u0924\u0915 \u091c\u094b \u0926\u0947\u0916\u093e \u0917\u092f\u093e \u0939\u0948, \u0909\u0938\u0938\u0947 \u0939\u092e\u0947\u0902 \u092f\u0939 \u0915\u093e\u092b\u0940 \u0938\u094d\u092a\u0937\u094d\u091f \u0939\u094b \u0917\u092f\u093e \u0939\u0948 \u0915\u093f \u090f\u0915 \u0905\u0935\u0915\u0932 \u0938\u092e\u0940\u0915\u0930\u0923 \u0915\u094d\u092f\u093e \u0939\u094b\u0924\u093e \u0939\u0948 \u0914\u0930 \u0907\u0938\u0915\u0947 \u0905\u0928\u0947\u0915 \u0909\u092a\u092f\u094b\u0917 \u0915\u094d\u092f\u093e \u0939\u094b \u0938\u0915\u0924\u0947 \u0939\u0948\u0902\u0964 \u0905\u092c \u0939\u092e \u0915\u0941\u091b \u092a\u0930\u093f\u092d\u093e\u0937\u093e\u0913\u0902 \u0914\u0930 \u0917\u0941\u0923\u094b\u0902 \u0915\u093e \u0905\u0927\u094d\u092f\u092f\u0928 \u0915\u0930\u0947\u0902\u0917\u0947 \u0924\u093e\u0915\u093f \u0907\u0938 \u0905\u0927\u094d\u092f\u092f\u0928 \u0915\u094b \u0906\u0917\u0947 \u0932\u0947 \u091c\u093e\u0928\u0947 \u0915\u0947 \u0932\u093f\u090f \u090f\u0915 \u092e\u091c\u092c\u0942\u0924 \u0938\u093e\u092e\u093e\u0928\u094d\u092f \u0906\u0927\u093e\u0930 \u0938\u094d\u0925\u093e\u092a\u093f\u0924 \u0915\u093f\u092f\u093e \u091c\u093e \u0938\u0915\u0947\u0964<\/p>\n<p><a name=\"LaEcuacionDiferencialOrdinariaDeOrdenK\"><\/a><\/p>\n<h3>\u0915\u094d\u0930\u092e k \u0915\u093e EDO<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=zE29azRIKng&#038;t=163s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #ff0000;\">\u090f\u0915 \u0938\u093e\u0927\u093e\u0930\u0923 \u0905\u0935\u0915\u0932 \u0938\u092e\u0940\u0915\u0930\u0923 (EDO)<\/span><\/strong><\/a> \u090f\u0915 \u0910\u0938\u093e \u0938\u092e\u0940\u0915\u0930\u0923 \u0939\u0948 \u091c\u093f\u0938\u092e\u0947\u0902 \u090f\u0915 \u0938\u094d\u0935\u0924\u0902\u0924\u094d\u0930 \u091a\u0930 <span class=\"katex-eq\" data-katex-display=\"false\">x<\/span>, \u090f\u0915 \u092b\u0932\u0928 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">y(x)<\/span><\/span>, \u0914\u0930 \u0907\u0938\u0915\u0947 \u0915\u0941\u091b \u0938\u093e\u0927\u093e\u0930\u0923 \u0905\u0935\u0915\u0932\u091c \u0936\u093e\u092e\u093f\u0932 \u0939\u094b\u0924\u0947 \u0939\u0948\u0902\u0964 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">y(x)<\/span><\/span> \u0915\u0947 \u092a\u0939\u0932\u0947 \u0915\u094d\u0930\u092e \u0915\u0947 \u0938\u093e\u0927\u093e\u0930\u0923 \u0905\u0935\u0915\u0932\u091c \u0915\u094b <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\frac{dy(x)}{dx}<\/span><\/span> \u092f\u093e <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">y&#039;(x)<\/span><\/span> \u0915\u0947 \u0930\u0942\u092a \u092e\u0947\u0902 \u0926\u0930\u094d\u0936\u093e\u092f\u093e \u091c\u093e\u0924\u093e \u0939\u0948, \u0926\u0942\u0938\u0930\u0947 \u0915\u094d\u0930\u092e \u0915\u0947 \u0915\u094b <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\frac{d^2y(x)}{dx^2}<\/span><\/span> \u092f\u093e <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">y&#039;&#039;(x)<\/span><\/span> \u0915\u0947 \u0930\u0942\u092a \u092e\u0947\u0902, \u0914\u0930 \u0938\u093e\u092e\u093e\u0928\u094d\u092f\u0924\u0903, \u0915\u094d\u0930\u092e <span class=\"katex-eq\" data-katex-display=\"false\">n<\/span> \u0915\u0947 \u0915\u094b <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\frac{d^ny(x)}{dx^n}<\/span><\/span> \u092f\u093e <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">y^{(n)}(x)<\/span><\/span> \u0915\u0947 \u0930\u0942\u092a \u092e\u0947\u0902 \u0926\u0930\u094d\u0936\u093e\u092f\u093e \u091c\u093e\u0924\u093e \u0939\u0948\u0964 \u0909\u0928 \u092e\u093e\u0928\u094b\u0902 \u0915\u093e \u0905\u0927\u093f\u0915\u0924\u092e <span class=\"katex-eq\" data-katex-display=\"false\">k<\/span>, \u091c\u093f\u0928\u0915\u0947 \u0932\u093f\u090f <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">y^{(k)}(x)<\/span><\/span> \u0938\u092e\u0940\u0915\u0930\u0923 \u092e\u0947\u0902 \u0906\u0924\u093e \u0939\u0948, \u0915\u094b \u0939\u092e <strong>\u0938\u092e\u0940\u0915\u0930\u0923 \u0915\u093e \u0915\u094d\u0930\u092e<\/strong> \u0915\u0939\u0924\u0947 \u0939\u0948\u0902\u0964 \u0907\u0938 \u092a\u094d\u0930\u0915\u093e\u0930, <strong>\u0915\u094d\u0930\u092e <span class=\"katex-eq\" data-katex-display=\"false\">k<\/span> \u0915\u0947 EDO \u0915\u093e \u0938\u093e\u092e\u093e\u0928\u094d\u092f \u0930\u0942\u092a<\/strong> \u0907\u0938 \u092a\u094d\u0930\u0915\u093e\u0930 \u0939\u0948:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F\\left(x,y(x),y&#039;(x), \\cdots, y^{(k)}(x)\\right)=0.<\/span>\n<p>\u0915\u0939\u093e \u091c\u093e\u0924\u093e \u0939\u0948 \u0915\u093f \u092f\u0926\u093f \u0915\u093f\u0938\u0940 \u0915\u094d\u0930\u092e <span class=\"katex-eq\" data-katex-display=\"false\">k<\/span> \u0915\u0940 EDO \u0915\u094b \u0907\u0938 \u092a\u094d\u0930\u0915\u093e\u0930 \u0935\u094d\u092f\u0915\u094d\u0924 \u0915\u093f\u092f\u093e \u091c\u093e\u090f \u0915\u093f <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">y^{(k)}(x)<\/span><\/span> \u0905\u0932\u0917 \u0915\u0930 \u0932\u093f\u092f\u093e \u091c\u093e\u090f, \u0924\u094b \u0935\u0939 <strong>\u0938\u093e\u092e\u093e\u0928\u094d\u092f \u0930\u0942\u092a<\/strong> \u092e\u0947\u0902 \u0939\u0948, \u0905\u0930\u094d\u0925\u093e\u0924\u094d:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">y^{(k)}(x) = f\\left(x,y(x),y&#039;(x), \\cdots, y^{(k-1)}(x)\\right).<\/span>\n<p>\u0938\u093e\u092e\u093e\u0928\u094d\u092f\u0924\u0903, \u092b\u0932\u0928 <span class=\"katex-eq\" data-katex-display=\"false\">y<\/span> \u090f\u0915 \u092b\u0932\u0928 \u0939\u094b\u0924\u093e \u0939\u0948 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R} \\longrightarrow \\mathbb{R}^n,<\/span><\/span> \u0907\u0938 \u092a\u094d\u0930\u0915\u093e\u0930 \u0915\u093f \u092f\u0939 \u0914\u0930 \u0907\u0938\u0915\u0947 \u0938\u092d\u0940 \u0905\u0935\u0915\u0932\u091c \u0915\u093f\u0938\u0940 \u092c\u093f\u0902\u0926\u0941 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x\\in\\mathbb{R}<\/span><\/span> \u092a\u0930 \u092e\u0942\u0932\u094d\u092f\u093e\u0902\u0915\u0928 \u0915\u093f\u090f \u091c\u093e\u0928\u0947 \u092a\u0930 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n<\/span><\/span> \u0915\u0947 \u0938\u0926\u093f\u0936 \u0939\u094b\u0924\u0947 \u0939\u0948\u0902\u0964 \u0907\u0938\u0947 \u0927\u094d\u092f\u093e\u0928 \u092e\u0947\u0902 \u0930\u0916\u0924\u0947 \u0939\u0941\u090f, \u092f\u0939 \u0938\u094d\u092a\u0937\u094d\u091f \u0939\u094b\u0924\u093e \u0939\u0948 \u0915\u093f, \u091a\u0942\u0902\u0915\u093f <span class=\"katex-eq\" data-katex-display=\"false\">F<\/span> \u091c\u094b EDO \u0915\u094b \u092a\u0930\u093f\u092d\u093e\u0937\u093f\u0924 \u0915\u0930\u0924\u093e \u0939\u0948 \u0909\u0938\u092e\u0947\u0902 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">1+(k+1)<\/span><\/span> \u091a\u0930\u094b\u0902 \u0939\u094b\u0924\u0947 \u0939\u0948\u0902, \u0907\u0938\u0932\u093f\u090f <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\text{Dom}(F)\\subset \\mathbb{R}^{1+n(k+1)}<\/span><\/span> \u0914\u0930 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\text{Rec}(F)\\subset \\mathbb{R}<\/span><\/span>; \u0914\u0930 \u0907\u0938\u0940 \u092a\u094d\u0930\u0915\u093e\u0930, <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\text{Dom}(f) = \\mathbb{R}^{1+nk}<\/span><\/span> \u0914\u0930 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\text{Rec}(f)\\subset \\mathbb{R}^n<\/span><\/span> \u0939\u094b\u0924\u093e \u0939\u0948\u0964<\/p>\n<p>\u0915\u094d\u0930\u092e <span class=\"katex-eq\" data-katex-display=\"false\">k<\/span> \u0915\u0947 EDO \u0915\u0947 \u0938\u093e\u092e\u093e\u0928\u094d\u092f \u0930\u0942\u092a \u0938\u0947 \u0909\u0938\u0915\u0947 \u0938\u093e\u092e\u093e\u0928\u094d\u092f \u0930\u0942\u092a \u092e\u0947\u0902 \u092a\u0930\u093f\u0935\u0930\u094d\u0924\u0928 <strong>\u0905\u0902\u0924\u0930\u094d\u0928\u093f\u0939\u093f\u0924 \u092b\u0932\u0928 \u0915\u0947 \u092a\u094d\u0930\u092e\u0947\u092f<\/strong> \u0915\u0947 \u0915\u093e\u0930\u0923 \u0938\u0902\u092d\u0935 \u0939\u094b\u0924\u093e \u0939\u0948\u0964<\/p>\n<p><a name=\"TeoremaDeLaFuncionImplicita\"><\/a><\/p>\n<h4>\u0905\u0902\u0924\u0930\u094d\u0928\u093f\u0939\u093f\u0924 \u092b\u0932\u0928 \u0915\u093e \u092a\u094d\u0930\u092e\u0947\u092f<\/h4>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=zE29azRIKng&#038;t=887s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #ff0000;\">\u092e\u093e\u0928 \u0932\u0940\u091c\u093f\u090f <span class=\"katex-eq\" data-katex-display=\"false\">F<\/span> \u090f\u0915 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathcal{C}^1<\/span><\/span> \u0935\u0930\u094d\u0917 \u0915\u093e \u092b\u0932\u0928 \u0939\u0948 \u090f\u0915 \u0916\u0941\u0932\u0947 \u0938\u092e\u0941\u091a\u094d\u091a\u092f <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">U \\subset \\mathbb{R}^n<\/span><\/span> \u092a\u0930<\/span><\/strong><\/a> \u091c\u093f\u0938\u0915\u093e \u092e\u0942\u0932\u094d\u092f \u0935\u093e\u0938\u094d\u0924\u0935\u093f\u0915 \u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902 \u092e\u0947\u0902 \u0939\u0948\u0964 \u0914\u0930 \u092e\u093e\u0928 \u0932\u0947\u0902 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(a_1,\\cdots, a_n) \\in U<\/span><\/span> \u0910\u0938\u093e \u0939\u0948 \u0915\u093f <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F(a_1,\\cdots, a_n) = 0<\/span><\/span> \u0914\u0930<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{\\partial F(a_1,\\cdots, a_n)}{\\partial x_n} \\neq 0<\/span>\n<p>\u0924\u092c \u090f\u0915 \u092a\u0921\u093c\u094b\u0938 <span class=\"katex-eq\" data-katex-display=\"false\">V<\/span> \u092e\u094c\u091c\u0942\u0926 \u0939\u0948 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(a_1, \\cdots, a_{n-1}) \\in \\mathbb{R}^{n-1}<\/span><\/span> \u0915\u093e \u0914\u0930 \u090f\u0915 \u092b\u0932\u0928 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\varphi:V \\longrightarrow \\mathbb{R}<\/span><\/span> \u0910\u0938\u093e \u0915\u093f:<\/p>\n<ol>\n<li type=\"i\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">V \\times \\varphi(V) \\subset U<\/span><\/span><\/li>\n<li type=\"i\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F(x_1,\\cdots,x_{n-1},x_n) = 0 \\leftrightarrow x_n = \\varphi(x_1,\\cdots, x_{n-1})<\/span><\/span><\/li>\n<li type=\"i\"><span class=\"katex-eq\" data-katex-display=\"false\">\\varphi<\/span> \u0905\u0935\u0915\u0932\u0928\u0940\u092f \u0939\u0948 \u0914\u0930\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle\\dfrac{\\partial \\varphi (a_1,\\cdots, a_{n-1})}{\\partial x_i} = - \\dfrac{ \\dfrac{\\partial F (a_1,\\cdots, a_n)}{\\partial x_i} }{ \\dfrac{\\partial F (a_1,\\cdots, a_n)}{\\partial x_n} }<\/span>\n<\/li>\n<\/ol>\n<h4>\u0905\u0902\u0924\u0930\u094d\u0928\u093f\u0939\u093f\u0924 \u092b\u0932\u0928 \u092a\u094d\u0930\u092e\u0947\u092f \u0915\u093e \u092a\u094d\u0930\u092e\u093e\u0923<\/h4>\n<h5>\u091c\u0948\u0915\u094b\u092c\u093f\u092f\u0928 \u092e\u0948\u091f\u094d\u0930\u093f\u0915\u094d\u0938 \u0938\u0947 \u0935\u094d\u092f\u0941\u0924\u094d\u092a\u0928\u094d\u0928<\/h5>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=zE29azRIKng&#038;t=1101s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #ff0000;\">\u092e\u093e\u0928 \u0932\u0940\u091c\u093f\u090f <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\psi(x_1,\\cdots,x_{n-1}, x_n) = (x_1,\\cdots,x_{n-1}, F(x_1,\\cdots, x_n)).<\/span><\/span><\/span><\/strong><\/a> \u092f\u0926\u093f \u0939\u092e \u0907\u0938\u0915\u0940 \u091c\u0948\u0915\u094b\u092c\u093f\u092f\u0928 \u092e\u0948\u091f\u094d\u0930\u093f\u0915\u094d\u0938 \u0915\u0940 \u0917\u0923\u0928\u093e \u0915\u0930\u0947\u0902, \u091c\u094b \u0928\u0940\u091a\u0947 \u0926\u093f\u0916\u093e\u0908 \u0917\u0908 \u0939\u0948:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\left( \\dfrac{\\partial \\psi(x_1,\\cdots, x_n)}{\\partial(x_1,\\cdots, x_n)} \\right) = \\left( \\begin{array}{cccc}\n\n1 &amp; 0 &amp;  \\cdots &amp; 0 \\\\\n\n0 &amp; 1 &amp;  \\cdots &amp; \\vdots \\\\\n\n\\vdots &amp;\\vdots &amp; \\ddots  &amp; \\vdots  \\\\\n\n\\displaystyle \\dfrac{\\partial F(x_1, \\cdots, x_n)}{\\partial x_1} &amp; \\dfrac{\\partial F(x_1, \\cdots, x_n)}{\\partial x_2} &amp; \\cdots  &amp; \\dfrac{\\partial F(x_1, \\cdots, x_n)}{\\partial x_n}\n\n\\end{array}\\right), <\/span>\n<p>\u0924\u094b \u0939\u092e \u0926\u0947\u0916\u0947\u0902\u0917\u0947 \u0915\u093f \u0907\u0938\u0915\u093e \u0921\u093f\u091f\u0930\u094d\u092e\u093f\u0928\u0947\u0902\u091f <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(a_1,\\cdots, a_n)<\/span><\/span> \u092a\u0930 \u0936\u0942\u0928\u094d\u092f \u0928\u0939\u0940\u0902 \u0939\u0948, \u0935\u093f\u0936\u0947\u0937 \u0930\u0942\u092a \u0938\u0947 \u0915\u094d\u092f\u094b\u0902\u0915\u093f, \u091c\u0948\u0938\u093e \u0915\u093f \u092a\u094d\u0930\u093e\u0930\u0902\u092d \u092e\u0947\u0902 \u0915\u0939\u093e \u0917\u092f\u093e \u0925\u093e, <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\partial F(a_1,\\cdots, a_n)\/\\partial x_n \\neq 0.<\/span><\/span> \u0907\u0938\u0938\u0947 \u0939\u092e \u0915\u0939 \u0938\u0915\u0924\u0947 \u0939\u0948\u0902 \u0915\u093f <span class=\"katex-eq\" data-katex-display=\"false\">\\psi<\/span> \u0915\u093e \u090f\u0915 \u0935\u094d\u092f\u0941\u0924\u094d\u0915\u094d\u0930\u092e \u0905\u0938\u094d\u0924\u093f\u0924\u094d\u0935 \u092e\u0947\u0902 \u0939\u0948 \u090f\u0915 \u0916\u0941\u0932\u0947 \u0938\u092e\u0941\u091a\u094d\u091a\u092f <span class=\"katex-eq\" data-katex-display=\"false\">W<\/span> \u092a\u0930 \u091c\u094b <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(a_1,\\cdots, a_n)<\/span><\/span> \u0915\u094b \u0938\u092e\u094d\u092e\u093f\u0932\u093f\u0924 \u0915\u0930\u0924\u093e \u0939\u0948\u0964<\/p>\n<h5>\u0939\u0932 \u0915\u093e \u0935\u093f\u0915\u093e\u0938<\/h5>\n<p>\u0905\u092c, \u090f\u0915 \u0938\u092e\u0941\u091a\u094d\u091a\u092f \u092a\u0930 \u0935\u093f\u091a\u093e\u0930 \u0915\u0930\u0947\u0902<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\tilde{V}=\\psi(W)\\ni \\psi(a_1,\\cdots,a_{n}) = (a_1,\\cdots,a_{n-1},F(a_1,\\cdots,a_{n}))=(a_1,\\cdots,a_{n-1},0).<\/span>\n<p>\u0907\u0938\u0938\u0947, \u0939\u092e \u090f\u0915 \u0905\u0928\u094d\u092f \u0938\u092e\u0941\u091a\u094d\u091a\u092f \u0915\u094b \u092a\u0930\u093f\u092d\u093e\u0937\u093f\u0924 \u0915\u0930 \u0938\u0915\u0924\u0947 \u0939\u0948\u0902<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">V=\\{(x_1,\\cdots,x_{n-1}) \\;|\\; (x_1,\\cdots,x_{n-1},0)\\in \\tilde{V}\\}\\ni (a_1,\\cdots,a_{n-1})<\/span>\n<p>\u0938\u092e\u0941\u091a\u094d\u091a\u092f <span class=\"katex-eq\" data-katex-display=\"false\">V<\/span> \u0907\u0938 \u092a\u094d\u0930\u0915\u093e\u0930, \u090f\u0915 \u0916\u0941\u0932\u093e \u0938\u092e\u0941\u091a\u094d\u091a\u092f \u0939\u0948 \u091c\u094b <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(a_1,\\cdots,a_{n-1})\\in\\mathbb{R}^{n-1}<\/span><\/span> \u0915\u094b \u0938\u092e\u094d\u092e\u093f\u0932\u093f\u0924 \u0915\u0930\u0924\u093e \u0939\u0948\u0964<\/p>\n<p>\u0907\u0938\u0915\u0947 \u0905\u0932\u093e\u0935\u093e, \u091a\u0942\u0902\u0915\u093f <span class=\"katex-eq\" data-katex-display=\"false\">\\psi<\/span> \u0915\u093e \u0935\u094d\u092f\u0941\u0924\u094d\u0915\u094d\u0930\u092e \u0905\u0938\u094d\u0924\u093f\u0924\u094d\u0935 \u092e\u0947\u0902 \u0939\u0948 (<span class=\"katex-eq\" data-katex-display=\"false\">W<\/span> \u092e\u0947\u0902), \u0924\u094b \u090f\u0915 \u0905\u0926\u094d\u0935\u093f\u0924\u0940\u092f <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(y_1,\\cdots,y_n)\\in W<\/span><\/span> \u092e\u094c\u091c\u0942\u0926 \u0939\u0948 \u0910\u0938\u093e \u0915\u093f <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\psi(y_1,\\cdots,y_n) = (x_1,\\cdots,x_{n-1},0).<\/span><\/span> \u0907\u0938\u0915\u093e \u0905\u0930\u094d\u0925 \u0939\u0948 \u0915\u093f:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rl} y_1 &amp;= x_1 \\\\ \\\\ \\vdots &amp; \\vdots \\\\ \\\\ y_{n-1} &amp;= x_{n-1} \\\\ \\\\ F(x_1,\\cdots,x_{n-1},y_n) &amp;= 0 \\end{array}<\/span>\n<p>\u0907\u0938 \u092a\u094d\u0930\u0915\u093e\u0930, \u0939\u092e \u092a\u0930\u093f\u092d\u093e\u0937\u093f\u0924 \u0915\u0930 \u0938\u0915\u0924\u0947 \u0939\u0948\u0902 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\varphi(x_1,\\cdots,x_{n-1}) = y_n<\/span><\/span>, \u091c\u093f\u0938\u0938\u0947 \u0915\u093f:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\psi^{-1}(x_1,\\cdots,x_{n-1},0) = (x_1,\\cdots,x_{n-1},\\varphi(x_1,\\cdots,x_{n-1}))<\/span>\n<p>\u0914\u0930<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F(x_1,\\cdots,x_{n-1},\\varphi(x_1,\\cdots,x_{n-1})) = 0<\/span>\n<p>\u0907\u0938\u0938\u0947 \u0939\u092e\u093e\u0930\u0947 \u092a\u093e\u0938 \u0939\u0948 \u0915\u093f <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\varphi(V)\\ni a_n,<\/span><\/span> \u0914\u0930 \u092a\u0930\u093f\u0923\u093e\u092e\u0938\u094d\u0935\u0930\u0942\u092a <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">V\\times\\varphi(V) \\subset U,<\/span><\/span> \u0914\u0930 \u0938\u093e\u0925 \u0939\u0940:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F(x_1,\\cdots,x_{n-1},x_n) = 0 \\leftrightarrow x_n = \\varphi(x_1,\\cdots,x_{n-1})<\/span>\n<h5>\u0905\u0935\u0915\u0932\u0928\u0940\u092f\u0924\u093e<\/h5>\n<p>\u0914\u0930 \u0905\u0902\u0924 \u092e\u0947\u0902, <span class=\"katex-eq\" data-katex-display=\"false\">\\psi<\/span> \u0915\u0940 \u0905\u0935\u0915\u0932\u0928\u0940\u092f\u0924\u093e <span class=\"katex-eq\" data-katex-display=\"false\">\\psi^{-1}<\/span> \u0915\u0940 \u0905\u0935\u0915\u0932\u0928\u0940\u092f\u0924\u093e \u0915\u0940 \u0913\u0930 \u0932\u0947 \u091c\u093e\u0924\u0940 \u0939\u0948, \u091c\u094b \u092c\u0926\u0932\u0947 \u092e\u0947\u0902 <span class=\"katex-eq\" data-katex-display=\"false\">\\varphi<\/span> \u0915\u0940 \u0905\u0935\u0915\u0932\u0928\u0940\u092f\u0924\u093e \u0915\u0940 \u0913\u0930 \u0932\u0947 \u091c\u093e\u0924\u0940 \u0939\u0948 <span class=\"katex-eq\" data-katex-display=\"false\">V<\/span> \u092a\u0930\u0964 \u0907\u0938\u0947 \u0927\u094d\u092f\u093e\u0928 \u092e\u0947\u0902 \u0930\u0916\u0924\u0947 \u0939\u0941\u090f, \u0939\u092e \u090f\u0915 \u092b\u0932\u0928 <span class=\"katex-eq\" data-katex-display=\"false\">g<\/span> \u0915\u094b \u0928\u093f\u092e\u094d\u0928\u0932\u093f\u0916\u093f\u0924 \u0938\u0902\u092c\u0902\u0927 \u0915\u0947 \u092e\u093e\u0927\u094d\u092f\u092e \u0938\u0947 \u092a\u0930\u093f\u092d\u093e\u0937\u093f\u0924 \u0915\u0930 \u0938\u0915\u0924\u0947 \u0939\u0948\u0902:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">g(x_1, \\cdots,x_{n-1}) = F(x_1,\\cdots,x_{n-1},\\varphi(x_1,\\cdots,x_{n-1})) = 0<\/span>\n<p>\u0914\u0930 \u092b\u093f\u0930, \u0936\u094d\u0930\u0943\u0902\u0916\u0932\u093e \u0928\u093f\u092f\u092e \u0915\u093e \u0909\u092a\u092f\u094b\u0917 \u0915\u0930\u0924\u0947 \u0939\u0941\u090f, \u0939\u092e\u093e\u0930\u0947 \u092a\u093e\u0938 \u0939\u0948:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{\\partial g}{\\partial x_i} = \\frac{\\partial F}{\\partial x_i} + \\frac{\\partial F}{\\partial x_n}\\frac{\\partial \\varphi }{\\partial x_i} = 0,<\/span>\n<p>\u091c\u0939\u093e\u0901 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">i=1,\\cdots, n-1.<\/span><\/span> \u0907\u0938 \u0905\u0902\u0924\u093f\u092e \u0938\u092e\u0940\u0915\u0930\u0923 \u0938\u0947 \u0939\u092e\u0947\u0902 \u092a\u094d\u0930\u093e\u092a\u094d\u0924 \u0939\u094b\u0924\u093e \u0939\u0948:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\dfrac{\\partial \\varphi(a_1,\\cdots,a_{n-1})}{\\partial x_i} = - \\dfrac{\\dfrac{\\partial F(a_1,\\cdots,a_{n})}{\\partial x_i}}{\\dfrac{\\partial F(a_1,\\cdots,a_{n})}{\\partial x_n}}<\/span>\n<p>\u0914\u0930 \u0907\u0938\u0915\u0947 \u0938\u093e\u0925 \u0939\u0940 \u0939\u092e \u0935\u0939 \u0938\u092c \u0938\u093f\u0926\u094d\u0927 \u0915\u0930 \u0932\u0947\u0924\u0947 \u0939\u0948\u0902 \u091c\u094b \u0938\u093f\u0926\u094d\u0927 \u0915\u0930\u0928\u093e \u0925\u093e \u25a0<\/p>\n<p><a name=\"LaSolucionDeUnaEcuacionDiferencialOrdinaria\"><\/a><\/p>\n<h3>\u090f\u0915 \u0938\u093e\u0927\u093e\u0930\u0923 \u0905\u0935\u0915\u0932 \u0938\u092e\u0940\u0915\u0930\u0923 \u0915\u093e \u0939\u0932<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=zE29azRIKng&#038;t=2249s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #ff0000;\">\u0906\u0907\u090f \u090f\u0915 \u0938\u093e\u092e\u093e\u0928\u094d\u092f \u0930\u0942\u092a \u092e\u0947\u0902 \u0935\u094d\u092f\u0915\u094d\u0924 EDO \u092a\u0930 \u0935\u093f\u091a\u093e\u0930 \u0915\u0930\u0947\u0902<\/span><\/strong><\/a><\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">y^{(n)} = f(x,y(x),y^\\prime(x),\\cdots,y^{(n-1)(x)})<\/span>\n<p>\u0924\u092c, \u090f\u0915 \u092b\u0932\u0928 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\varphi : I_\\phi \\longmapsto \\mathbb{R}^n,<\/span><\/span> \u091c\u0939\u093e\u0901 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">I_\\phi<\/span><\/span> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}<\/span><\/span> \u0915\u093e \u090f\u0915 \u0905\u0902\u0924\u0930\u093e\u0932 \u0939\u0948, \u0915\u094b <strong>EDO \u0915\u093e \u090f\u0915 \u0939\u0932<\/strong> \u0915\u0939\u093e \u091c\u093e\u0924\u093e \u0939\u0948 \u092f\u0926\u093f:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\left(\\forall x \\in I_\\phi \\right) \\left(\\varphi^{(n)}(x) = f(x,\\varphi(x),\\varphi^\\prime(x),\\cdots,\\varphi^{(n-1)(x)}\\right)<\/span>\n<p><a name=\"CuidadoConElDominioDeDefinicionDeLasSoluciones\"><\/a><\/p>\n<h4>\u0939\u0932 \u0915\u0947 \u092a\u0930\u093f\u092d\u093e\u0937\u093e \u0915\u094d\u0937\u0947\u0924\u094d\u0930 \u0915\u0947 \u092a\u094d\u0930\u0924\u093f \u0938\u093e\u0935\u0927\u093e\u0928\u0940<\/h4>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=zE29azRIKng&#038;t=2387s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #ff0000;\">\u0907\u0938 \u092c\u093f\u0902\u0926\u0941 \u092a\u0930 \u092f\u0939 \u0906\u0935\u0936\u094d\u092f\u0915 \u0939\u0948<\/span><\/strong><\/a> \u0915\u093f \u0905\u0935\u0915\u0932 \u0938\u092e\u0940\u0915\u0930\u0923 \u0915\u0947 \u0939\u0932 \u0915\u0947 \u092a\u0930\u093f\u092d\u093e\u0937\u093e \u0915\u094d\u0937\u0947\u0924\u094d\u0930 \u0915\u094b \u0938\u094d\u092a\u0937\u094d\u091f \u0930\u0942\u092a \u0938\u0947 \u0918\u094b\u0937\u093f\u0924 \u0915\u0930\u0928\u0947 \u0915\u0947 \u092e\u0939\u0924\u094d\u0935 \u092a\u0930 \u092c\u0932 \u0926\u093f\u092f\u093e \u091c\u093e\u090f\u0964 \u0909\u0926\u093e\u0939\u0930\u0923 \u0915\u0947 \u0932\u093f\u090f, \u0909\u0938 \u092b\u0932\u0928 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\phi<\/span><\/span> \u0915\u093e \u092a\u0930\u093f\u092d\u093e\u0937\u093e \u0915\u094d\u0937\u0947\u0924\u094d\u0930, \u091c\u093f\u0938\u0915\u093e \u0909\u0932\u094d\u0932\u0947\u0916 \u092a\u093f\u091b\u0932\u0947 \u0905\u0928\u0941\u091a\u094d\u091b\u0947\u0926 \u092e\u0947\u0902 \u0915\u093f\u092f\u093e \u0917\u092f\u093e \u0925\u093e, <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">I_\\phi<\/span><\/span> \u0905\u0902\u0924\u0930\u093e\u0932 \u0939\u0948\u0964 \u092f\u0939 \u092e\u0939\u0924\u094d\u0935\u092a\u0942\u0930\u094d\u0923 \u0939\u0948 \u0915\u094d\u092f\u094b\u0902\u0915\u093f \u0905\u0935\u0915\u0932 \u0938\u092e\u0940\u0915\u0930\u0923\u094b\u0902 \u092a\u0930 \u0915\u093e\u0930\u094d\u092f \u0915\u0930\u0924\u0947 \u0938\u092e\u092f \u090f\u0915 \u0938\u093e\u092e\u093e\u0928\u094d\u092f \u0924\u094d\u0930\u0941\u091f\u093f \u092f\u0939 \u092e\u093e\u0928 \u0932\u0947\u0928\u093e \u0939\u094b\u0924\u093e \u0939\u0948 \u0915\u093f \u0926\u094b \u0939\u0932 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\phi_1<\/span><\/span> \u0914\u0930 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\phi_2<\/span><\/span> \u0938\u092e\u093e\u0928 \u0939\u0948\u0902 \u0915\u0947\u0935\u0932 \u0907\u0938\u0932\u093f\u090f \u0915\u093f <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\left(\\forall x \\in I_{\\phi_1}\\cap I_{\\phi_2}\\right)\\left(\\phi_1(x) = \\phi_2(x)\\right),<\/span><\/span> \u091c\u092c\u0915\u093f \u0935\u093e\u0938\u094d\u0924\u0935 \u092e\u0947\u0902 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">I_{\\phi_1}\\neq I_{\\phi_2}.<\/span><\/span> \u0907\u0938 \u092c\u093f\u0902\u0926\u0941 \u0915\u094b \u0938\u094d\u092a\u0937\u094d\u091f \u0915\u0930\u0928\u0947 \u0915\u0947 \u0932\u093f\u090f, \u0906\u0907\u090f \u0928\u093f\u092e\u094d\u0928\u0932\u093f\u0916\u093f\u0924 \u0905\u0935\u0915\u0932 \u0938\u092e\u0940\u0915\u0930\u0923 \u0915\u093e \u0905\u0927\u094d\u092f\u092f\u0928 \u0915\u0930\u0947\u0902:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">y^\\prime = -y^2.<\/span>\n<p>\u0907\u0938 EDO \u0915\u093e \u090f\u0915 \u0938\u0902\u092d\u093e\u0935\u093f\u0924 \u0939\u0932 \u092b\u0932\u0928 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\psi_1 : ]0,+\\infty[ \\longrightarrow \\mathbb{R}^+\\setminus\\{0\\}<\/span><\/span> \u0939\u0948, \u091c\u093f\u0938\u0947 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\psi_1(x)=1\/x<\/span><\/span> \u0915\u0947 \u0930\u0942\u092a \u092e\u0947\u0902 \u092a\u0930\u093f\u092d\u093e\u0937\u093f\u0924 \u0915\u093f\u092f\u093e \u0917\u092f\u093e \u0939\u0948, \u0915\u094d\u092f\u094b\u0902\u0915\u093f <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\psi_1^{\\prime} = -1\/x^2 = -\\psi_1^2<\/span><\/span> \u0915\u093f\u0938\u0940 \u092d\u0940 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x\\in]0,+\\infty[<\/span><\/span> \u0915\u0947 \u0932\u093f\u090f \u0939\u094b\u0924\u093e \u0939\u0948\u0964 \u0932\u0947\u0915\u093f\u0928 \u0915\u0941\u091b \u092c\u0940\u091c\u0917\u0923\u093f\u0924\u0940\u092f \u091a\u093e\u0932\u094b\u0902 \u0915\u0947 \u092e\u093e\u0927\u094d\u092f\u092e \u0938\u0947, \u092f\u0926\u093f \u0939\u092e \u0935\u093f\u0935\u0930\u0923\u094b\u0902 \u092a\u0930 \u0927\u094d\u092f\u093e\u0928 \u0928 \u0926\u0947\u0902 \u0924\u094b \u0939\u092e \u0907\u0938\u0938\u0947 \u090f\u0915 \u092a\u0942\u0930\u0940 \u0924\u0930\u0939 \u092d\u093f\u0928\u094d\u0928 \u0939\u0932 \u092a\u0930 \u092a\u0939\u0941\u0901\u091a \u0938\u0915\u0924\u0947 \u0939\u0948\u0902\u0964 \u0909\u0926\u093e\u0939\u0930\u0923 \u0915\u0947 \u0932\u093f\u090f, \u092f\u0939 \u0938\u094d\u092a\u0937\u094d\u091f \u0939\u0948 \u0915\u093f:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{1}{x} = \\frac{1}{1 - (1-x)},<\/span>\n<p>\u0914\u0930 \u0907\u0938 \u0938\u092e\u0924\u093e \u0915\u093e \u0926\u093e\u092f\u093e\u0901 \u092a\u0915\u094d\u0937 \u0928\u093f\u092e\u094d\u0928\u0932\u093f\u0916\u093f\u0924 \u091c\u094d\u092f\u093e\u092e\u093f\u0924\u0940\u092f \u0936\u094d\u0930\u0947\u0923\u0940 \u0915\u093e \u092a\u0930\u093f\u0923\u093e\u092e \u0939\u0948:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\"> \\displaystyle \\sum_{n=0}^{+\\infty} (1-x)^n = \\frac{1}{1 - (1-x)}<\/span>\n<p>\u0907\u0938 \u092a\u094d\u0930\u0915\u093e\u0930, \u0907\u0928 \u0917\u0942\u0922\u093c \u0915\u0932\u093e\u0913\u0902 \u092e\u0947\u0902 \u0915\u092e \u0905\u0928\u0941\u092d\u0935\u0940 \u0915\u094b\u0908 \u0935\u094d\u092f\u0915\u094d\u0924\u093f \u092f\u0939 \u092e\u093e\u0928 \u0938\u0915\u0924\u093e \u0939\u0948 \u0915\u093f \u092b\u0932\u0928 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\psi_1<\/span><\/span> \u0914\u0930 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\psi_2 = \\sum_{n=0}^{+\\infty} (1-x)^n <\/span><\/span> \u0939\u092e\u0947\u0902 \u0935\u0939\u0940 \u0939\u0932 \u092a\u094d\u0930\u0926\u093e\u0928 \u0915\u0930\u0924\u0947 \u0939\u0948\u0902 \u091c\u094b \u090a\u092a\u0930 \u0926\u0940 \u0917\u0908 \u0905\u0935\u0915\u0932 \u0938\u092e\u0940\u0915\u0930\u0923 \u0915\u0947 \u0932\u093f\u090f \u0909\u092a\u092f\u0941\u0915\u094d\u0924 \u0939\u0948, \u0915\u094d\u092f\u094b\u0902\u0915\u093f \u0935\u093e\u0938\u094d\u0924\u0935 \u092e\u0947\u0902 \u0909\u0928\u0915\u0947 \u092a\u0930\u093f\u0923\u093e\u092e \u092e\u0947\u0932 \u0916\u093e\u0924\u0947 \u0939\u0948\u0902; \u0939\u093e\u0932\u093e\u0901\u0915\u093f, \u092f\u0939 \u0927\u094d\u092f\u093e\u0928 \u0928\u0939\u0940\u0902 \u0926\u093f\u092f\u093e \u091c\u093e\u090f\u0917\u093e \u0915\u093f \u092f\u0939 \u091c\u094d\u092f\u093e\u092e\u093f\u0924\u0940\u092f \u0936\u094d\u0930\u0947\u0923\u0940 \u0915\u0947\u0935\u0932 \u0924\u092c \u092e\u093e\u0928\u094d\u092f \u0939\u0948 \u091c\u092c <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">|1-x| \\lt 1<\/span><\/span>, \u0905\u0930\u094d\u0925\u093e\u0924 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x\\in]0,2[<\/span><\/span> \u0915\u0947 \u0932\u093f\u090f\u0964 \u0932\u0947\u0915\u093f\u0928 \u0914\u0930 \u092d\u0940 \u0939\u0948: \u091a\u0942\u0902\u0915\u093f <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">]0,2[\\subset]0,+\\infty[<\/span><\/span>, \u0907\u0938\u0932\u093f\u090f \u092f\u0939 \u092d\u0940 \u0938\u0924\u094d\u092f \u0939\u0948 \u0915\u093f <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\psi_1<\/span><\/span> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\psi_2<\/span><\/span> \u0915\u093e \u0935\u093f\u0938\u094d\u0924\u093e\u0930 \u0915\u0930\u0924\u093e \u0939\u0948, \u0915\u094d\u092f\u094b\u0902\u0915\u093f \u091c\u0939\u093e\u0901 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\psi_2<\/span><\/span> \u092a\u0930\u093f\u092d\u093e\u0937\u093f\u0924 \u0939\u0948, \u0935\u0939\u093e\u0901 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\psi_1<\/span><\/span> \u092d\u0940 \u092a\u0930\u093f\u092d\u093e\u0937\u093f\u0924 \u0939\u0948 \u2014 \u0914\u0930 \u0909\u0938\u0938\u0947 \u092d\u0940 \u0906\u0917\u0947\u0964<\/P><\/p>\n<p><a name=\"SolucionExtendidaYSolucionMaximal\"><\/a><\/p>\n<h4>\u0935\u093f\u0938\u094d\u0924\u093e\u0930\u093f\u0924 \u0939\u0932 \u0914\u0930 \u0905\u0927\u093f\u0915\u0924\u092e \u0939\u0932<\/h4>\n<p>\u0906\u0907\u090f \u0926\u094b \u092b\u0932\u0928\u094b\u0902 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\phi_1<\/span><\/span> \u0914\u0930 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\phi_2<\/span><\/span> \u092a\u0930 \u0935\u093f\u091a\u093e\u0930 \u0915\u0930\u0947\u0902, \u091c\u094b \u0915\u094d\u0930\u092e\u0936\u0903 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">I_{\\phi_1}<\/span><\/span> \u0914\u0930 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">I_{\\phi_2}<\/span><\/span> \u0905\u0902\u0924\u0930\u093e\u0932\u094b\u0902 \u092a\u0930 \u092a\u0930\u093f\u092d\u093e\u0937\u093f\u0924 \u0939\u0948\u0902, \u0914\u0930 \u091c\u094b \u0915\u093f\u0938\u0940 \u0905\u0935\u0915\u0932 \u0938\u092e\u0940\u0915\u0930\u0923 \u0915\u0947 \u0939\u0932 \u0939\u0948\u0902\u0964 \u092f\u0926\u093f <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">I_{\\phi_1}\\subset I_{\\phi_2},<\/span><\/span> \u0924\u094b \u0915\u0939\u093e \u091c\u093e\u0924\u093e \u0939\u0948 \u0915\u093f \u0939\u0932 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\phi_2<\/span><\/span> \u0939\u0932 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\phi_1<\/span><\/span> \u0915\u093e \u0935\u093f\u0938\u094d\u0924\u093e\u0930 \u0915\u0930\u0924\u093e \u0939\u0948, \u092f\u093e \u0915\u093f <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\phi_2<\/span><\/span> \u0939\u0932 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\phi_1<\/span><\/span> \u0915\u0940 \u0924\u0941\u0932\u0928\u093e \u092e\u0947\u0902 \u0905\u0927\u093f\u0915 \u0938\u093e\u092e\u093e\u0928\u094d\u092f \u0939\u0948\u0964 \u090f\u0915 \u0939\u0932 <span class=\"katex-eq\" data-katex-display=\"false\">\\phi<\/span> \u0915\u094b \u00ab\u0905\u0927\u093f\u0915\u0924\u092e\u00bb \u0915\u0939\u093e \u091c\u093e\u0924\u093e \u0939\u0948 \u092f\u0926\u093f \u0910\u0938\u093e \u0915\u094b\u0908 \u0905\u0928\u094d\u092f \u0939\u0932 \u0928 \u0939\u094b \u091c\u094b \u0909\u0938\u0947 \u0917\u0948\u0930-\u0924\u0941\u091a\u094d\u091b \u0930\u0942\u092a \u092e\u0947\u0902 \u0935\u093f\u0938\u094d\u0924\u093e\u0930 \u0915\u0930\u0947\u0964<\/p>\n<p><a name=\"SolucionExplicitaYSolucionImplicita\"><\/a><\/p>\n<h4>\u0938\u094d\u092a\u0937\u094d\u091f \u0939\u0932 \u0914\u0930 \u0905\u092a\u094d\u0930\u0924\u094d\u092f\u0915\u094d\u0937 \u0939\u0932<\/h4>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=zE29azRIKng&#038;t=2649s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #ff0000;\">\u090f\u0915 \u092b\u0932\u0928<\/span><\/strong><\/a> <span class=\"katex-eq\" data-katex-display=\"false\">\\phi<\/span> \u0915\u094b \u0915\u094d\u0930\u092e <span class=\"katex-eq\" data-katex-display=\"false\">n<\/span> \u0915\u0940 EDO (\u0938\u093e\u092e\u093e\u0928\u094d\u092f \u0930\u0942\u092a \u092e\u0947\u0902 \u0932\u093f\u0916\u0940 \u0917\u0908) \u0915\u093e \u0939\u0932 \u092e\u093e\u0928\u093e \u091c\u093e\u0924\u093e \u0939\u0948:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">y^{(n)}(x)=f(x,y(x),y^\\prime(x),\\cdots,y^{(n-1)}(x)),<\/span>\n<p> \u0915\u093f\u0938\u0940 \u0905\u0902\u0924\u0930\u093e\u0932 <span class=\"katex-eq\" data-katex-display=\"false\">I<\/span> \u092e\u0947\u0902 \u092f\u0926\u093f<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(\\forall x\\in I)\\left(\\phi^{n}(x) = f(x,\\phi(x),\\phi^\\prime(x),\\cdots,\\phi^{(n-1)}(x))\\right)<\/span>\n<p>\u091c\u094b \u0939\u092e\u0928\u0947 \u092a\u0939\u0932\u0947 \u0915\u0908 \u0905\u0928\u0941\u091a\u094d\u091b\u0947\u0926\u094b\u0902 \u092e\u0947\u0902 \u0926\u0947\u0916\u093e, \u0935\u0939\u0940 \u0939\u0948 \u091c\u093f\u0938\u0947 <strong>\u0905\u0902\u0924\u0930\u093e\u0932 <span class=\"katex-eq\" data-katex-display=\"false\">I<\/span> \u092e\u0947\u0902 \u0905\u0935\u0915\u0932 \u0938\u092e\u0940\u0915\u0930\u0923 \u0915\u093e \u0938\u094d\u092a\u0937\u094d\u091f \u0939\u0932<\/strong> \u0915\u0939\u093e \u091c\u093e\u0924\u093e \u0939\u0948\u0964 \u091c\u0948\u0938\u093e \u0915\u093f \u0928\u093e\u092e \u0938\u0947 \u0938\u094d\u092a\u0937\u094d\u091f \u0939\u0948, \u0939\u0932\u094b\u0902 \u0915\u094b \u092a\u0930\u093f\u092d\u093e\u0937\u093f\u0924 \u0915\u0930\u0928\u0947 \u0915\u093e \u090f\u0915 \u0905\u092a\u094d\u0930\u0924\u094d\u092f\u0915\u094d\u0937 \u0930\u0942\u092a \u092d\u0940 \u0939\u094b\u0924\u093e \u0939\u0948\u0964 \u0915\u0939\u093e \u091c\u093e\u0924\u093e \u0939\u0948 \u0915\u093f \u0915\u094b\u0908 \u0938\u0902\u092c\u0902\u0927 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\Phi(x,y)=0<\/span><\/span> <strong>\u0905\u0902\u0924\u0930\u093e\u0932 <span class=\"katex-eq\" data-katex-display=\"false\">I<\/span> \u092e\u0947\u0902 \u0905\u0935\u0915\u0932 \u0938\u092e\u0940\u0915\u0930\u0923 \u0915\u093e \u0905\u092a\u094d\u0930\u0924\u094d\u092f\u0915\u094d\u0937 \u0939\u0932<\/strong> \u0939\u0948 \u092f\u0926\u093f \u092f\u0939 <span class=\"katex-eq\" data-katex-display=\"false\">I<\/span> \u092e\u0947\u0902 \u0926\u094b \u092f\u093e \u0905\u0927\u093f\u0915 \u0905\u092a\u094d\u0930\u0924\u094d\u092f\u0915\u094d\u0937 \u0939\u0932\u094b\u0902 \u0915\u094b \u092a\u0930\u093f\u092d\u093e\u0937\u093f\u0924 \u0915\u0930\u0924\u093e \u0939\u0948\u0964<\/p>\n<h3>\u0928\u093f\u0937\u094d\u0915\u0930\u094d\u0937<\/h3>\n<p>\u0907\u0938 \u0915\u0915\u094d\u0937\u093e \u092e\u0947\u0902 \u0939\u092e\u0928\u0947 \u0938\u093e\u0927\u093e\u0930\u0923 \u0905\u0935\u0915\u0932 \u0938\u092e\u0940\u0915\u0930\u0923 \u0915\u0940 \u0938\u0902\u0915\u0932\u094d\u092a\u0928\u093e \u0915\u094b \u090f\u0915 \u0915\u0920\u094b\u0930 \u0932\u0947\u0915\u093f\u0928 \u0938\u0941\u0932\u092d \u0926\u0943\u0937\u094d\u091f\u093f\u0915\u094b\u0923 \u0938\u0947 \u0935\u093f\u092d\u093e\u091c\u093f\u0924 \u0915\u093f\u092f\u093e \u0939\u0948, \u0914\u0930 \u0909\u0928 \u0914\u092a\u091a\u093e\u0930\u093f\u0915 \u0928\u0940\u0902\u0935\u094b\u0902 \u0915\u0940 \u0938\u094d\u0925\u093e\u092a\u0928\u093e \u0915\u0940 \u0939\u0948 \u091c\u094b \u0939\u092e\u0947\u0902 \u0928 \u0915\u0947\u0935\u0932 \u090f\u0915 EDO \u0915\u094b \u092a\u0939\u091a\u093e\u0928\u0928\u0947 \u0915\u0940 \u0905\u0928\u0941\u092e\u0924\u093f \u0926\u0947\u0924\u0940 \u0939\u0948\u0902, \u092c\u0932\u094d\u0915\u093f \u0909\u0928\u0915\u0947 \u0939\u0932\u094b\u0902 \u0915\u0947 \u092a\u0940\u091b\u0947 \u0915\u0940 \u0924\u0930\u094d\u0915\u0936\u0915\u094d\u0924\u093f \u0915\u094b \u092d\u0940 \u0938\u092e\u091d\u0928\u0947 \u092e\u0947\u0902 \u0938\u0915\u094d\u0937\u092e \u092c\u0928\u093e\u0924\u0940 \u0939\u0948\u0902\u0964 \u0905\u0902\u0924\u0930\u094d\u0928\u093f\u0939\u093f\u0924 \u092b\u0932\u0928 \u0915\u0947 \u092a\u094d\u0930\u092e\u0947\u092f \u0915\u0940 \u092c\u0926\u094c\u0932\u0924, \u0939\u092e \u0938\u093e\u092e\u093e\u0928\u094d\u092f \u0930\u0942\u092a \u0914\u0930 \u0938\u093e\u092e\u093e\u0928\u094d\u092f\u0940\u0915\u0943\u0924 \u0930\u0942\u092a \u0915\u0947 \u092c\u0940\u091a \u092a\u0930\u093f\u0935\u0930\u094d\u0924\u0928 \u0915\u094b \u0938\u094d\u092a\u0937\u094d\u091f \u0930\u0942\u092a \u0938\u0947 \u0909\u091a\u093f\u0924 \u0920\u0939\u0930\u093e \u092a\u093e\u090f, \u091c\u094b \u0935\u094d\u092f\u093e\u0935\u0939\u093e\u0930\u093f\u0915 \u0938\u092e\u0938\u094d\u092f\u093e\u0913\u0902 \u0938\u0947 \u0928\u093f\u092a\u091f\u0928\u0947 \u0915\u0947 \u0932\u093f\u090f \u090f\u0915 \u092e\u0939\u0924\u094d\u0935\u092a\u0942\u0930\u094d\u0923 \u0924\u0915\u0928\u0940\u0915\u0940 \u0915\u094d\u0937\u092e\u0924\u093e \u0939\u0948\u0964<\/p>\n<p>\u0907\u0938\u0915\u0947 \u0905\u0932\u093e\u0935\u093e, \u0939\u092e\u0928\u0947 \u0938\u091f\u0940\u0915 \u0930\u0942\u092a \u0938\u0947 \u0909\u0928 \u0935\u093f\u092d\u093f\u0928\u094d\u0928 \u0924\u0930\u0940\u0915\u094b\u0902 \u0915\u094b \u092d\u0940 \u092a\u0939\u091a\u093e\u0928\u093e \u091c\u093f\u0928\u092e\u0947\u0902 \u0915\u093f\u0938\u0940 \u0939\u0932 \u0915\u094b \u0938\u092e\u091d\u093e \u091c\u093e \u0938\u0915\u0924\u093e \u0939\u0948: \u0938\u094d\u092a\u0937\u094d\u091f \u092f\u093e \u0905\u092a\u094d\u0930\u0924\u094d\u092f\u0915\u094d\u0937, \u0935\u093f\u0938\u094d\u0924\u093e\u0930\u093f\u0924 \u092f\u093e \u0905\u0927\u093f\u0915\u0924\u092e; \u0914\u0930 \u0939\u092e\u0928\u0947 \u0907\u0938 \u092c\u093e\u0924 \u092a\u0930 \u092c\u0932 \u0926\u093f\u092f\u093e \u2014 \u091c\u093f\u0938\u0947 \u0905\u0915\u094d\u0938\u0930 \u0915\u092e \u0906\u0902\u0915\u093e \u091c\u093e\u0924\u093e \u0939\u0948 \u2014 \u0915\u093f \u0909\u0928\u0915\u0947 \u092a\u0930\u093f\u092d\u093e\u0937\u093e \u0915\u094d\u0937\u0947\u0924\u094d\u0930 \u0915\u094b \u0909\u091a\u093f\u0924 \u0930\u0942\u092a \u0938\u0947 \u0918\u094b\u0937\u093f\u0924 \u0915\u0930\u0928\u093e \u0915\u093f\u0924\u0928\u093e \u092e\u0939\u0924\u094d\u0935\u092a\u0942\u0930\u094d\u0923 \u0939\u0948\u0964 \u092f\u0947 \u092d\u0947\u0926 \u0915\u0947\u0935\u0932 \u0914\u092a\u091a\u093e\u0930\u093f\u0915 \u0928\u0939\u0940\u0902 \u0939\u0948\u0902: \u092f\u0947 \u0915\u094d\u0930\u093f\u092f\u093e\u0924\u094d\u092e\u0915 \u0939\u0948\u0902\u0964 \u091c\u0948\u0938\u093e \u0915\u093f \u0939\u092e\u0928\u0947 \u0926\u0947\u0916\u093e, \u0907\u0928\u0915\u0940 \u0909\u092a\u0947\u0915\u094d\u0937\u093e \u0915\u0930\u0928\u0947 \u0938\u0947 \u092a\u094d\u0930\u093e\u092a\u094d\u0924 \u092a\u0930\u093f\u0923\u093e\u092e\u094b\u0902 \u0915\u0940 \u0935\u094d\u092f\u093e\u0916\u094d\u092f\u093e \u092e\u0947\u0902 \u0917\u0902\u092d\u0940\u0930 \u0935\u0948\u091a\u093e\u0930\u093f\u0915 \u0924\u094d\u0930\u0941\u091f\u093f\u092f\u093e\u0901 \u0939\u094b \u0938\u0915\u0924\u0940 \u0939\u0948\u0902\u0964<\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u0938\u093e\u0927\u093e\u0930\u0923 \u0905\u0935\u0915\u0932 \u0938\u092e\u0940\u0915\u0930\u0923 (EDO) \u0915\u094d\u092f\u093e \u0939\u0948? \u0938\u093e\u0930\u093e\u0902\u0936:\u0907\u0938 \u0915\u0915\u094d\u0937\u093e \u092e\u0947\u0902, \u0915\u094d\u0930\u092e k \u0915\u0947 \u0938\u093e\u0927\u093e\u0930\u0923 \u0905\u0935\u0915\u0932 \u0938\u092e\u0940\u0915\u0930\u0923\u094b\u0902 (EDO) \u0915\u093e \u0905\u0928\u094d\u0935\u0947\u0937\u0923 \u0915\u093f\u092f\u093e \u0917\u092f\u093e \u0939\u0948, \u0907\u0938\u0915\u0940 \u092a\u0930\u093f\u092d\u093e\u0937\u093e \u0914\u0930 \u0938\u093e\u092e\u093e\u0928\u094d\u092f \u090f\u0935\u0902 \u092e\u093e\u0928\u0915 \u0930\u0942\u092a \u092e\u0947\u0902 \u0909\u0938\u0915\u0940 \u0905\u092d\u093f\u0935\u094d\u092f\u0915\u094d\u0924\u093f \u0938\u0947 \u0936\u0941\u0930\u0941\u0906\u0924 \u0915\u0930\u0924\u0947 \u0939\u0941\u090f\u0964 \u091c\u0948\u0915\u094b\u092c\u093f\u092f\u0928 \u092e\u0948\u091f\u094d\u0930\u093f\u0915\u094d\u0938 \u0914\u0930 \u0905\u0902\u0924\u0930\u094d\u0928\u093f\u0939\u093f\u0924 \u092b\u0932\u0928 \u092a\u094d\u0930\u092e\u0947\u092f \u091c\u0948\u0938\u0947 \u0905\u0935\u0927\u093e\u0930\u0923\u093e\u0913\u0902 \u0915\u0947 \u092e\u093e\u0927\u094d\u092f\u092e \u0938\u0947, \u0907\u0928 \u0938\u092e\u0940\u0915\u0930\u0923\u094b\u0902 \u0915\u0947 \u0939\u0932 \u0914\u0930 \u0938\u0902\u092c\u0926\u094d\u0927 \u0917\u0941\u0923\u094b\u0902 \u0915\u094b \u0938\u092e\u091d\u0928\u0947 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