{"id":32858,"date":"2025-04-03T23:02:39","date_gmt":"2025-04-03T23:02:39","guid":{"rendered":"http:\/\/toposuranos.com\/material\/?p=32858"},"modified":"2025-04-03T23:02:39","modified_gmt":"2025-04-03T23:02:39","slug":"%d9%85%d8%a7-%d9%87%d9%8a-%d8%a7%d9%84%d9%85%d8%b9%d8%a7%d8%af%d9%84%d8%a9-%d8%a7%d9%84%d8%aa%d9%81%d8%a7%d8%b6%d9%84%d9%8a%d8%a9-%d8%a7%d9%84%d8%b9%d8%a7%d8%af%d9%8a%d8%a9-edo%d8%9f","status":"publish","type":"post","link":"http:\/\/toposuranos.com\/material\/ar\/%d9%85%d8%a7-%d9%87%d9%8a-%d8%a7%d9%84%d9%85%d8%b9%d8%a7%d8%af%d9%84%d8%a9-%d8%a7%d9%84%d8%aa%d9%81%d8%a7%d8%b6%d9%84%d9%8a%d8%a9-%d8%a7%d9%84%d8%b9%d8%a7%d8%af%d9%8a%d8%a9-edo%d8%9f\/","title":{"rendered":"\u0645\u0627 \u0647\u064a \u0627\u0644\u0645\u0639\u0627\u062f\u0644\u0629 \u0627\u0644\u062a\u0641\u0627\u0636\u0644\u064a\u0629 \u0627\u0644\u0639\u0627\u062f\u064a\u0629 (EDO)\u061f"},"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>\u0645\u0627 \u0647\u064a \u0627\u0644\u0645\u0639\u0627\u062f\u0644\u0629 \u0627\u0644\u062a\u0641\u0627\u0636\u0644\u064a\u0629 \u0627\u0644\u0639\u0627\u062f\u064a\u0629 (EDO)\u061f<\/h1>\n<p style=\"text-align:center;\" dir=\"rtl\"><em><strong>\u0627\u0644\u0645\u0644\u062e\u0635:<\/strong><\/br>\u0641\u064a \u0647\u0630\u0647 \u0627\u0644\u062d\u0635\u0629\u060c \u064a\u062a\u0645 \u0627\u0633\u062a\u0643\u0634\u0627\u0641 \u0627\u0644\u0645\u0639\u0627\u062f\u0644\u0627\u062a \u0627\u0644\u062a\u0641\u0627\u0636\u0644\u064a\u0629 \u0627\u0644\u0639\u0627\u062f\u064a\u0629 (EDO) \u0645\u0646 \u0627\u0644\u0631\u062a\u0628\u0629 k\u060c \u0628\u062f\u0621\u064b\u0627 \u0645\u0646 \u062a\u0639\u0631\u064a\u0641\u0647\u0627 \u0648\u062a\u0645\u062b\u064a\u0644\u0647\u0627 \u0628\u0627\u0644\u0634\u0643\u0644\u064a\u0646 \u0627\u0644\u0639\u0627\u062f\u064a \u0648\u0627\u0644\u0639\u0627\u0645. \u0645\u0646 \u062e\u0644\u0627\u0644 \u0645\u0641\u0627\u0647\u064a\u0645 \u0645\u062b\u0644 \u0645\u0635\u0641\u0648\u0641\u0629 \u064a\u0639\u0642\u0648\u0628\u064a \u0648\u0645\u0628\u0631\u0647\u0646\u0629 \u0627\u0644\u062f\u0627\u0644\u0629 \u0627\u0644\u0636\u0645\u0646\u064a\u0629\u060c \u064a\u062a\u0645 \u0648\u0636\u0639 \u0627\u0644\u0623\u0633\u0627\u0633 \u0644\u0641\u0647\u0645 \u062d\u0644\u0648\u0644 \u0647\u0630\u0647 \u0627\u0644\u0645\u0639\u0627\u062f\u0644\u0627\u062a \u0648\u0627\u0644\u062e\u0635\u0627\u0626\u0635 \u0627\u0644\u0645\u0631\u062a\u0628\u0637\u0629 \u0628\u0647\u0627\u060c \u0645\u062b\u0644 \u0645\u062c\u0627\u0644 \u0627\u0644\u062a\u0639\u0631\u064a\u0641 \u0648\u0627\u0644\u062d\u0644\u0648\u0644 \u0627\u0644\u0635\u0631\u064a\u062d\u0629 \u0648\u0627\u0644\u0636\u0645\u0646\u064a\u0629.<\/em><\/p>\n<p style=\"text-align:center\"><strong>\u0623\u0647\u0640\u0640\u062f\u0627\u0641 \u0627\u0644\u062a\u0639\u0644\u0640\u0640\u0645<\/strong><\/p>\n<p>\u0639\u0646\u062f \u0646\u0647\u0627\u064a\u0629 \u0647\u0630\u0647 \u0627\u0644\u062d\u0635\u0629 \u0633\u064a\u0643\u0648\u0646 \u0627\u0644\u0637\u0627\u0644\u0628 \u0642\u0627\u062f\u0631\u064b\u0627 \u0639\u0644\u0649:<\/p>\n<ol>\n<li><strong>\u0627\u0633\u062a\u0630\u0643\u0627\u0631<\/strong> \u062a\u0639\u0631\u064a\u0641 \u0648\u062e\u0635\u0627\u0626\u0635 \u0627\u0644\u0645\u0639\u0627\u062f\u0644\u0629 \u0627\u0644\u062a\u0641\u0627\u0636\u0644\u064a\u0629 \u0627\u0644\u0639\u0627\u062f\u064a\u0629 (EDO) \u0627\u0644\u0623\u0633\u0627\u0633\u064a\u0629.<\/li>\n<li><strong>\u0634\u0631\u062d<\/strong> \u0627\u0644\u0639\u0644\u0627\u0642\u0629 \u0628\u064a\u0646 \u0627\u0644\u0645\u0639\u0627\u062f\u0644\u0629 \u0627\u0644\u062a\u0641\u0627\u0636\u0644\u064a\u0629 \u0627\u0644\u0639\u0627\u062f\u064a\u0629 \u0648\u062d\u0644\u0648\u0644\u0647\u0627 \u0627\u0644\u0645\u0645\u0643\u0646\u0629.<\/li>\n<\/ol>\n<p style=\"text-align:center;\" dir=\"rtl\"><strong>\u0627\u0644\u0641\u0647\u0631\u0633<\/strong><br \/>\n<a href=\"#LaEcuacionDiferencialOrdinariaDeOrdenK\"><strong>\u0627\u0644\u0645\u0639\u0627\u062f\u0644\u0629 \u0627\u0644\u062a\u0641\u0627\u0636\u0644\u064a\u0629 \u0627\u0644\u0639\u0627\u062f\u064a\u0629 \u0645\u0646 \u0627\u0644\u0631\u062a\u0628\u0629 k<\/strong><\/a><br \/>\n<a href=\"#TeoremaDeLaFuncionImplicita\">\u0645\u0628\u0631\u0647\u0646\u0629 \u0627\u0644\u062f\u0627\u0644\u0629 \u0627\u0644\u0636\u0645\u0646\u064a\u0629<\/a><br \/>\n<a href=\"#LaSolucionDeUnaEcuacionDiferencialOrdinaria\"><strong>\u062d\u0644 \u0627\u0644\u0645\u0639\u0627\u062f\u0644\u0629 \u0627\u0644\u062a\u0641\u0627\u0636\u0644\u064a\u0629 \u0627\u0644\u0639\u0627\u062f\u064a\u0629<\/strong><\/a><br \/>\n<a href=\"#CuidadoConElDominioDeDefinicionDeLasSoluciones\">\u062a\u0646\u0628\u064a\u0647 \u0628\u062e\u0635\u0648\u0635 \u0645\u062c\u0627\u0644 \u062a\u0639\u0631\u064a\u0641 \u0627\u0644\u062d\u0644\u0648\u0644<\/a><br \/>\n<a href=\"#SolucionExtendidaYSolucionMaximal\">\u0627\u0644\u062d\u0644 \u0627\u0644\u0645\u0645\u062a\u062f \u0648\u0627\u0644\u062d\u0644 \u0627\u0644\u0623\u0642\u0635\u0649<\/a><br \/>\n<a href=\"#SolucionExplicitaYSolucionImplicita\">\u0627\u0644\u062d\u0644 \u0627\u0644\u0635\u0631\u064a\u062d \u0648\u0627\u0644\u062d\u0644 \u0627\u0644\u0636\u0645\u0646\u064a<\/a>\n<\/p>\n<p><center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/zE29azRIKng\" title=\"\u0645\u0634\u063a\u0644 \u0641\u064a\u062f\u064a\u0648 \u064a\u0648\u062a\u064a\u0648\u0628\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen><\/iframe><\/center><\/p>\n<p>\u0628\u0645\u0627 \u0631\u0623\u064a\u0646\u0627\u0647 \u062d\u062a\u0649 \u0627\u0644\u0622\u0646\u060c \u0623\u0635\u0628\u062d \u0644\u062f\u064a\u0646\u0627 \u0641\u0643\u0631\u0629 \u0648\u0627\u0636\u062d\u0629 \u0625\u0644\u0649 \u062d\u062f \u0643\u0628\u064a\u0631 \u062d\u0648\u0644 \u0645\u0627\u0647\u064a\u0629 \u0627\u0644\u0645\u0639\u0627\u062f\u0644\u0629 \u0627\u0644\u062a\u0641\u0627\u0636\u0644\u064a\u0629 \u0648\u0627\u0644\u062a\u0637\u0628\u064a\u0642\u0627\u062a \u0627\u0644\u0645\u062a\u0639\u062f\u062f\u0629 \u0627\u0644\u062a\u064a \u064a\u0645\u0643\u0646 \u0623\u0646 \u062a\u0643\u0648\u0646 \u0644\u0647\u0627. \u0633\u0646\u062a\u0648\u0642\u0641 \u0627\u0644\u0622\u0646 \u0644\u062f\u0631\u0627\u0633\u0629 \u0628\u0639\u0636 \u0627\u0644\u062a\u0639\u0627\u0631\u064a\u0641 \u0648\u0627\u0644\u062e\u0635\u0627\u0626\u0635 \u0628\u0647\u062f\u0641 \u062a\u0623\u0633\u064a\u0633 \u0642\u0627\u0639\u062f\u0629 \u0645\u0634\u062a\u0631\u0643\u0629 \u0642\u0648\u064a\u0629 \u0644\u0645\u0648\u0627\u0635\u0644\u0629 \u0647\u0630\u0627 \u0627\u0644\u062f\u0631\u0633.<\/p>\n<p><a name=\"LaEcuacionDiferencialOrdinariaDeOrdenK\"><\/a><\/p>\n<h3>\u0627\u0644\u0645\u0639\u0627\u062f\u0644\u0629 \u0627\u0644\u062a\u0641\u0627\u0636\u0644\u064a\u0629 \u0627\u0644\u0639\u0627\u062f\u064a\u0629 \u0645\u0646 \u0627\u0644\u0631\u062a\u0628\u0629 k<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=zE29azRIKng&#038;t=163s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #ff0000;\">\u0627\u0644\u0645\u0639\u0627\u062f\u0644\u0629 \u0627\u0644\u062a\u0641\u0627\u0636\u0644\u064a\u0629 \u0627\u0644\u0639\u0627\u062f\u064a\u0629 (EDO)<\/span><\/strong><\/a> \u0647\u064a \u0645\u0639\u0627\u062f\u0644\u0629 \u062a\u062a\u0636\u0645\u0646 \u0645\u062a\u063a\u064a\u0631\u064b\u0627 \u0645\u0633\u062a\u0642\u0644\u0627\u064b <span class=\"katex-eq\" data-katex-display=\"false\">x<\/span>\u060c \u0648\u062f\u0627\u0644\u0629 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">y(x)<\/span><\/span>\u060c \u0648\u0628\u0639\u0636 \u0645\u0634\u062a\u0642\u0627\u062a\u0647\u0627 \u0627\u0644\u0639\u0627\u062f\u064a\u0629. \u062a\u064f\u0643\u062a\u0628 \u0627\u0644\u0645\u0634\u062a\u0642\u0627\u062a \u0627\u0644\u0639\u0627\u062f\u064a\u0629 \u0645\u0646 \u0627\u0644\u0631\u062a\u0628\u0629 \u0627\u0644\u0623\u0648\u0644\u0649 \u0644\u0640 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">y(x)<\/span><\/span> \u0628\u0627\u0633\u062a\u062e\u062f\u0627\u0645 \u0631\u0645\u0648\u0632 \u0645\u062b\u0644 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\frac{dy(x)}{dx}<\/span><\/span> \u0623\u0648 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">y&#039;(x)<\/span><\/span>\u060c \u0648\u0645\u0646 \u0627\u0644\u0631\u062a\u0628\u0629 \u0627\u0644\u062b\u0627\u0646\u064a\u0629 \u0645\u062b\u0644 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\frac{d^2y(x)}{dx^2}<\/span><\/span> \u0623\u0648 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">y&#039;&#039;(x)<\/span><\/span>\u060c \u0648\u0628\u0634\u0643\u0644 \u0639\u0627\u0645 \u0645\u0646 \u0627\u0644\u0631\u062a\u0628\u0629 <span class=\"katex-eq\" data-katex-display=\"false\">n<\/span> \u0645\u062b\u0644 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\frac{d^ny(x)}{dx^n}<\/span><\/span> \u0623\u0648 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">y^{(n)}(x)<\/span><\/span>. \u0625\u0646 \u0627\u0644\u0642\u064a\u0645\u0629 \u0627\u0644\u0639\u0638\u0645\u0649 \u0644\u0640 <span class=\"katex-eq\" data-katex-display=\"false\">k<\/span> \u0627\u0644\u062a\u064a \u064a\u0638\u0647\u0631 \u0641\u064a\u0647\u0627 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">y^{(k)}(x)<\/span><\/span> \u0641\u064a \u0627\u0644\u0645\u0639\u0627\u062f\u0644\u0629 \u062a\u064f\u0633\u0645\u0649 <strong>\u0631\u062a\u0628\u0629 \u0627\u0644\u0645\u0639\u0627\u062f\u0644\u0629<\/strong>. \u0648\u0628\u0647\u0630\u0627\u060c \u0641\u0625\u0646 <strong>\u0627\u0644\u0634\u0643\u0644 \u0627\u0644\u0639\u0627\u0645 \u0644\u0645\u0639\u0627\u062f\u0644\u0629 \u062a\u0641\u0627\u0636\u0644\u064a\u0629 \u0639\u0627\u062f\u064a\u0629 \u0645\u0646 \u0627\u0644\u0631\u062a\u0628\u0629 <span class=\"katex-eq\" data-katex-display=\"false\">k<\/span><\/strong> \u0647\u0648:<\/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>\u064a\u064f\u0642\u0627\u0644 \u0625\u0646 \u0627\u0644\u0645\u0639\u0627\u062f\u0644\u0629 \u0627\u0644\u062a\u0641\u0627\u0636\u0644\u064a\u0629 \u0627\u0644\u0639\u0627\u062f\u064a\u0629 \u0645\u0646 \u0627\u0644\u0631\u062a\u0628\u0629 <span class=\"katex-eq\" data-katex-display=\"false\">k<\/span> \u0641\u064a <strong>\u0627\u0644\u0634\u0643\u0644 \u0627\u0644\u0637\u0628\u064a\u0639\u064a<\/strong> \u0625\u0630\u0627 \u062a\u0645\u062a \u0635\u064a\u0627\u063a\u062a\u0647\u0627 \u0628\u062d\u064a\u062b \u062a\u0643\u0648\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">y^{(k)}(x)<\/span><\/span> \u0645\u0639\u0632\u0648\u0644\u0629 \u0641\u064a \u0637\u0631\u0641 \u0627\u0644\u0645\u0639\u0627\u062f\u0644\u0629\u060c \u0623\u064a:<\/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>\u0628\u0634\u0643\u0644 \u0639\u0627\u0645\u060c \u062a\u0643\u0648\u0646 \u0627\u0644\u062f\u0627\u0644\u0629 <span class=\"katex-eq\" data-katex-display=\"false\">y<\/span> \u062f\u0627\u0644\u0629 \u0645\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R} \\longrightarrow \\mathbb{R}^n,<\/span><\/span> \u0628\u062d\u064a\u062b \u062a\u0643\u0648\u0646 \u0647\u0630\u0647 \u0627\u0644\u062f\u0627\u0644\u0629 \u0648\u062c\u0645\u064a\u0639 \u0645\u0634\u062a\u0642\u0627\u062a\u0647\u0627 \u0639\u0646\u062f \u0623\u064a \u0646\u0642\u0637\u0629 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x\\in\\mathbb{R}<\/span><\/span> \u0639\u0628\u0627\u0631\u0629 \u0639\u0646 \u0645\u062a\u062c\u0647\u0627\u062a \u0641\u064a <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n<\/span><\/span>. \u0648\u0628\u0646\u0627\u0621\u064b \u0639\u0644\u0649 \u0630\u0644\u0643\u060c \u0648\u0644\u0623\u0646 \u0627\u0644\u062f\u0627\u0644\u0629 <span class=\"katex-eq\" data-katex-display=\"false\">F<\/span> \u0627\u0644\u062a\u064a \u062a\u0635\u0641 \u0627\u0644\u0645\u0639\u0627\u062f\u0644\u0629 \u0645\u0646 \u0627\u0644\u0631\u062a\u0628\u0629 <span class=\"katex-eq\" data-katex-display=\"false\">k<\/span> \u062a\u0639\u062a\u0645\u062f \u0639\u0644\u0649 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">1+(k+1)<\/span><\/span> \u0645\u062a\u063a\u064a\u0631\u064b\u0627\u060c \u0641\u0625\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\text{Dom}(F)\\subset \\mathbb{R}^{1+n(k+1)}<\/span><\/span> \u0648 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\text{Rec}(F)\\subset \\mathbb{R}<\/span><\/span>\u061b \u0648\u0628\u0634\u0643\u0644 \u0645\u0645\u0627\u062b\u0644\u060c <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\text{Dom}(f) = \\mathbb{R}^{1+nk}<\/span><\/span> \u0648 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\text{Rec}(f)\\subset \\mathbb{R}^n<\/span><\/span>.<\/p>\n<p>\u0625\u0646 \u0627\u0644\u0627\u0646\u062a\u0642\u0627\u0644 \u0645\u0646 \u0627\u0644\u0634\u0643\u0644 \u0627\u0644\u0639\u0627\u0645 \u0644\u0644\u0645\u0639\u0627\u062f\u0644\u0629 \u0627\u0644\u062a\u0641\u0627\u0636\u0644\u064a\u0629 \u0627\u0644\u0639\u0627\u062f\u064a\u0629 \u0645\u0646 \u0627\u0644\u0631\u062a\u0628\u0629 <span class=\"katex-eq\" data-katex-display=\"false\">k<\/span> \u0625\u0644\u0649 \u0634\u0643\u0644\u0647\u0627 \u0627\u0644\u0637\u0628\u064a\u0639\u064a \u0645\u0645\u0643\u0646 \u0628\u0641\u0636\u0644 <strong>\u0645\u0628\u0631\u0647\u0646\u0629 \u0627\u0644\u062f\u0627\u0644\u0629 \u0627\u0644\u0636\u0645\u0646\u064a\u0629.<\/strong><\/p>\n<p><a name=\"TeoremaDeLaFuncionImplicita\"><\/a><\/p>\n<h4>\u0645\u0628\u0631\u0647\u0646\u0629 \u0627\u0644\u062f\u0627\u0644\u0629 \u0627\u0644\u0636\u0645\u0646\u064a\u0629<\/h4>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=zE29azRIKng&#038;t=887s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #ff0000;\">\u0644\u062a\u0643\u0646 <span class=\"katex-eq\" data-katex-display=\"false\">F<\/span> \u062f\u0627\u0644\u0629 \u0645\u0646 \u0627\u0644\u0635\u0646\u0641 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathcal{C}^1<\/span><\/span> \u0639\u0644\u0649 \u0645\u062c\u0645\u0648\u0639\u0629 \u0645\u0641\u062a\u0648\u062d\u0629 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">U \\subset \\mathbb{R}^n<\/span><\/span><\/span><\/strong><\/a> \u0630\u0627\u062a \u0642\u064a\u0645 \u062d\u0642\u064a\u0642\u064a\u0629. \u0648\u0644\u0646\u0641\u0631\u0636 \u0623\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(a_1,\\cdots, a_n) \\in U<\/span><\/span> \u0628\u062d\u064a\u062b <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F(a_1,\\cdots, a_n) = 0<\/span><\/span> \u0648<\/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>\u0625\u0630\u064b\u0627\u060c \u064a\u0648\u062c\u062f \u062d\u064a\u0651 <span class=\"katex-eq\" data-katex-display=\"false\">V<\/span> \u0644\u0644\u0646\u0642\u0637\u0629 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(a_1, \\cdots, a_{n-1}) \\in \\mathbb{R}^{n-1}<\/span><\/span> \u0648\u062f\u0627\u0644\u0629 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\varphi:V \\longrightarrow \\mathbb{R}<\/span><\/span> \u062a\u062d\u0642\u0642 \u0645\u0627 \u064a\u0644\u064a:<\/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> \u0642\u0627\u0628\u0644\u0629 \u0644\u0644\u0627\u0634\u062a\u0642\u0627\u0642 \u0648\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>\u0628\u0631\u0647\u0627\u0646 \u0645\u0628\u0631\u0647\u0646\u0629 \u0627\u0644\u062f\u0627\u0644\u0629 \u0627\u0644\u0636\u0645\u0646\u064a\u0629<\/h4>\n<h5>\u062a\u0637\u0648\u064a\u0631 \u0627\u0646\u0637\u0644\u0627\u0642\u064b\u0627 \u0645\u0646 \u0645\u0635\u0641\u0648\u0641\u0629 \u064a\u0639\u0642\u0648\u0628\u064a<\/h5>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=zE29azRIKng&#038;t=1101s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #ff0000;\">\u0644\u062a\u0643\u0646 <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> \u0625\u0630\u0627 \u062d\u0633\u0628\u0646\u0627 \u0645\u0635\u0641\u0648\u0641\u0629 \u064a\u0639\u0642\u0648\u0628\u064a \u0627\u0644\u062e\u0627\u0635\u0629 \u0628\u0647\u0627\u060c \u0643\u0645\u0627 \u0647\u0648 \u0645\u0648\u0636\u062d \u0623\u062f\u0646\u0627\u0647:<\/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>\u0633\u0646\u0644\u0627\u062d\u0638 \u0623\u0646 \u0627\u0644\u0645\u062d\u062f\u062f \u0644\u0647\u0627 \u063a\u064a\u0631 \u0635\u0641\u0631\u064a \u0641\u064a <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(a_1,\\cdots, a_n)<\/span><\/span>\u060c \u0648\u0630\u0644\u0643 \u0628\u0627\u0644\u0636\u0628\u0637 \u0644\u0623\u0646\u0647\u060c \u0643\u0645\u0627 \u062a\u0645 \u0625\u062b\u0628\u0627\u062a\u0647 \u0641\u064a \u0627\u0644\u0628\u062f\u0627\u064a\u0629\u060c <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> \u0648\u0628\u0646\u0627\u0621\u064b \u0639\u0644\u0649 \u0630\u0644\u0643\u060c \u064a\u0645\u0643\u0646\u0646\u0627 \u0627\u0644\u0642\u0648\u0644 \u0625\u0646 <span class=\"katex-eq\" data-katex-display=\"false\">\\psi<\/span> \u062a\u0645\u062a\u0644\u0643 \u0645\u0639\u0643\u0648\u0633\u064b\u0627 \u0639\u0644\u0649 \u0645\u062c\u0645\u0648\u0639\u0629 \u0645\u0641\u062a\u0648\u062d\u0629 <span class=\"katex-eq\" data-katex-display=\"false\">W<\/span> \u062a\u062d\u062a\u0648\u064a \u0639\u0644\u0649 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(a_1,\\cdots, a_n).<\/span><\/span><\/p>\n<h5>\u062a\u0637\u0648\u064a\u0631 \u0627\u0644\u062d\u0644<\/h5>\n<p>\u0627\u0644\u0622\u0646\u060c \u0644\u0646\u0639\u062a\u0628\u0631 \u0645\u062c\u0645\u0648\u0639\u0629<\/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>\u0648\u0645\u0646 \u0647\u0646\u0627\u060c \u064a\u0645\u0643\u0646\u0646\u0627 \u062a\u0639\u0631\u064a\u0641 \u0645\u062c\u0645\u0648\u0639\u0629 \u0623\u062e\u0631\u0649<\/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>\u0625\u0646 \u0627\u0644\u0645\u062c\u0645\u0648\u0639\u0629 <span class=\"katex-eq\" data-katex-display=\"false\">V<\/span> \u0647\u064a \u0628\u0627\u0644\u062a\u0627\u0644\u064a \u0645\u062c\u0645\u0648\u0639\u0629 \u0645\u0641\u062a\u0648\u062d\u0629 \u062a\u062d\u062a\u0648\u064a \u0639\u0644\u0649 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(a_1,\\cdots,a_{n-1})\\in\\mathbb{R}^{n-1}.<\/span><\/span><\/p>\n<p>\u0648\u0628\u0645\u0627 \u0623\u0646 <span class=\"katex-eq\" data-katex-display=\"false\">\\psi<\/span> \u062a\u0645\u062a\u0644\u0643 \u0645\u0639\u0643\u0648\u0633\u064b\u0627 (\u0641\u064a <span class=\"katex-eq\" data-katex-display=\"false\">W<\/span>)\u060c \u0641\u0647\u0646\u0627\u0643 \u0639\u0646\u0635\u0631 \u0648\u062d\u064a\u062f <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(y_1,\\cdots,y_n)\\in W<\/span><\/span> \u064a\u062d\u0642\u0642 <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> \u0648\u0647\u0630\u0627 \u064a\u0639\u0646\u064a \u0623\u0646:<\/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>\u0648\u0628\u0627\u0644\u062a\u0627\u0644\u064a\u060c \u064a\u0645\u0643\u0646\u0646\u0627 \u062a\u0639\u0631\u064a\u0641 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\varphi(x_1,\\cdots,x_{n-1}) = y_n<\/span><\/span> \u0628\u062d\u064a\u062b:<\/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>\u0648<\/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>\u0648\u0628\u0627\u0644\u062a\u0627\u0644\u064a\u060c \u0644\u062f\u064a\u0646\u0627 \u0623\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\varphi(V)\\ni a_n,<\/span><\/span> \u0648\u0628\u0646\u0627\u0621\u064b \u0639\u0644\u064a\u0647 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">V\\times\\varphi(V) \\subset U,<\/span><\/span> \u0648\u0643\u0630\u0644\u0643:<\/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>\u0627\u0644\u0642\u0627\u0628\u0644\u064a\u0629 \u0644\u0644\u0627\u0634\u062a\u0642\u0627\u0642<\/h5>\n<p>\u0648\u0623\u062e\u064a\u0631\u064b\u0627\u060c \u0641\u0625\u0646 \u0642\u0627\u0628\u0644\u064a\u0629 <span class=\"katex-eq\" data-katex-display=\"false\">\\psi<\/span> \u0644\u0644\u0627\u0634\u062a\u0642\u0627\u0642 \u062a\u0624\u062f\u064a \u0625\u0644\u0649 \u0642\u0627\u0628\u0644\u064a\u0629 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\psi^{-1}<\/span><\/span> \u0644\u0644\u0627\u0634\u062a\u0642\u0627\u0642\u060c \u0645\u0645\u0627 \u064a\u0624\u062f\u064a \u0628\u062f\u0648\u0631\u0647 \u0625\u0644\u0649 \u0642\u0627\u0628\u0644\u064a\u0629 <span class=\"katex-eq\" data-katex-display=\"false\">\\varphi<\/span> \u0644\u0644\u0627\u0634\u062a\u0642\u0627\u0642 \u0639\u0644\u0649 <span class=\"katex-eq\" data-katex-display=\"false\">V<\/span>. \u0648\u0628\u0646\u0627\u0621\u064b \u0639\u0644\u0649 \u0630\u0644\u0643\u060c \u064a\u0645\u0643\u0646\u0646\u0627 \u062a\u0639\u0631\u064a\u0641 \u062f\u0627\u0644\u0629 <span class=\"katex-eq\" data-katex-display=\"false\">g<\/span> \u0645\u0646 \u062e\u0644\u0627\u0644 \u0627\u0644\u0639\u0644\u0627\u0642\u0629:<\/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>\u062b\u0645 \u0628\u0627\u0633\u062a\u062e\u062f\u0627\u0645 \u0642\u0627\u0639\u062f\u0629 \u0627\u0644\u0633\u0644\u0633\u0644\u0629\u060c \u0646\u062d\u0635\u0644 \u0639\u0644\u0649:<\/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>\u062d\u064a\u062b <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">i=1,\\cdots, n-1.<\/span><\/span> \u0648\u0645\u0646 \u0647\u0630\u0647 \u0627\u0644\u0645\u0639\u0627\u062f\u0644\u0629 \u0627\u0644\u0623\u062e\u064a\u0631\u0629 \u0646\u0633\u062a\u0646\u062a\u062c:<\/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>\u0648\u0628\u0630\u0644\u0643 \u0646\u0643\u0648\u0646 \u0642\u062f \u0623\u062b\u0628\u062a\u0646\u0627 \u0643\u0644 \u0645\u0627 \u0623\u0631\u062f\u0646\u0627 \u0625\u062b\u0628\u0627\u062a\u0647 \u25a0<\/p>\n<p><a name=\"LaSolucionDeUnaEcuacionDiferencialOrdinaria\"><\/a><\/p>\n<h3>\u062d\u0644 \u0627\u0644\u0645\u0639\u0627\u062f\u0644\u0629 \u0627\u0644\u062a\u0641\u0627\u0636\u0644\u064a\u0629 \u0627\u0644\u0639\u0627\u062f\u064a\u0629<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=zE29azRIKng&#038;t=2249s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #ff0000;\">\u0644\u0646\u0646\u0638\u0631 \u0641\u064a \u0645\u0639\u0627\u062f\u0644\u0629 \u062a\u0641\u0627\u0636\u0644\u064a\u0629 \u0639\u0627\u062f\u064a\u0629 \u0645\u0643\u062a\u0648\u0628\u0629 \u0628\u0627\u0644\u0634\u0643\u0644 \u0627\u0644\u0637\u0628\u064a\u0639\u064a<\/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>\u0625\u0630\u064b\u0627\u060c \u0641\u0625\u0646 \u062f\u0627\u0644\u0629 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\varphi : I_\\phi \\longmapsto \\mathbb{R}^n,<\/span><\/span> \u062d\u064a\u062b <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">I_\\phi<\/span><\/span> \u0647\u0648 \u0641\u062a\u0631\u0629 \u0636\u0645\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R},<\/span><\/span> \u062a\u064f\u0633\u0645\u0649 <strong>\u062d\u0644\u0627\u064b \u0644\u0644\u0645\u0639\u0627\u062f\u0644\u0629 \u0627\u0644\u062a\u0641\u0627\u0636\u0644\u064a\u0629 \u0627\u0644\u0639\u0627\u062f\u064a\u0629<\/strong> \u0625\u0630\u0627 \u062a\u062d\u0642\u0642:<\/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>\u0627\u0646\u062a\u0628\u0627\u0647 \u0625\u0644\u0649 \u0645\u062c\u0627\u0644 \u062a\u0639\u0631\u064a\u0641 \u0627\u0644\u062d\u0644\u0648\u0644<\/h4>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=zE29azRIKng&#038;t=2387s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #ff0000;\">\u0641\u064a \u0647\u0630\u0647 \u0627\u0644\u0645\u0631\u062d\u0644\u0629\u060c \u0645\u0646 \u0627\u0644\u0636\u0631\u0648\u0631\u064a \u0627\u0644\u062a\u0623\u0643\u064a\u062f<\/span><\/strong><\/a> \u0639\u0644\u0649 \u0623\u0647\u0645\u064a\u0629 \u0627\u0644\u062a\u0635\u0631\u064a\u062d \u0628\u0634\u0643\u0644 \u0635\u0631\u064a\u062d \u0639\u0646 \u0645\u062c\u0627\u0644 \u062a\u0639\u0631\u064a\u0641 \u062d\u0644 \u0627\u0644\u0645\u0639\u0627\u062f\u0644\u0629 \u0627\u0644\u062a\u0641\u0627\u0636\u0644\u064a\u0629. \u0639\u0644\u0649 \u0633\u0628\u064a\u0644 \u0627\u0644\u0645\u062b\u0627\u0644\u060c \u0641\u0625\u0646 \u0645\u062c\u0627\u0644 \u0627\u0644\u062f\u0627\u0644\u0629 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\phi<\/span><\/span> \u0627\u0644\u062a\u064a \u062a\u062d\u062f\u062b\u0646\u0627 \u0639\u0646\u0647\u0627 \u0641\u064a \u0627\u0644\u0641\u0642\u0631\u0629 \u0627\u0644\u0633\u0627\u0628\u0642\u0629 \u0647\u0648 \u0627\u0644\u0641\u062a\u0631\u0629 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">I_\\phi.<\/span><\/span> \u0648\u0647\u0630\u0627 \u0623\u0645\u0631 \u0645\u0647\u0645 \u0644\u0623\u0646 \u062e\u0637\u0623\u064b \u0634\u0627\u0626\u0639\u064b\u0627 \u0641\u064a \u0627\u0644\u062a\u0639\u0627\u0645\u0644 \u0645\u0639 \u0627\u0644\u0645\u0639\u0627\u062f\u0644\u0627\u062a \u0627\u0644\u062a\u0641\u0627\u0636\u0644\u064a\u0629 \u0647\u0648 \u0627\u0639\u062a\u0628\u0627\u0631 \u062d\u0644\u064a\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\phi_1<\/span><\/span> \u0648 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\phi_2<\/span><\/span> \u0645\u062a\u0633\u0627\u0648\u064a\u064a\u0646 \u0641\u0642\u0637 \u0644\u0623\u0646 <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> \u0628\u0627\u0644\u0631\u063a\u0645 \u0645\u0646 \u0623\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">I_{\\phi_1}\\neq I_{\\phi_2}.<\/span><\/span> \u0644\u0634\u0631\u062d \u0647\u0630\u0647 \u0627\u0644\u0646\u0642\u0637\u0629\u060c \u0644\u0646\u0641\u062d\u0635 \u0627\u0644\u0645\u0639\u0627\u062f\u0644\u0629 \u0627\u0644\u062a\u0641\u0627\u0636\u0644\u064a\u0629 \u0627\u0644\u062a\u0627\u0644\u064a\u0629:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">y^\\prime = -y^2.<\/span>\n<p>\u0623\u062d\u062f \u0627\u0644\u062d\u0644\u0648\u0644 \u0627\u0644\u0645\u0645\u0643\u0646\u0629 \u0644\u0647\u0630\u0647 \u0627\u0644\u0645\u0639\u0627\u062f\u0644\u0629 \u0647\u0648 \u0627\u0644\u062f\u0627\u0644\u0629 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\psi_1 : ]0,+\\infty[ \\longrightarrow \\mathbb{R}^+\\setminus\\{0\\}<\/span><\/span> \u0627\u0644\u0645\u0639\u0631\u0641\u0629 \u0639\u0644\u0649 \u0627\u0644\u0646\u062d\u0648 \u0627\u0644\u062a\u0627\u0644\u064a <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\psi_1(x)=1\/x,<\/span><\/span> \u0644\u0623\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\psi_1^{\\prime} = -1\/x^2 = -\\psi_1^2<\/span><\/span> \u0644\u0643\u0644 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x\\in]0,+\\infty[.<\/span><\/span> \u0648\u0644\u0643\u0646 \u0645\u0639 \u0628\u0639\u0636 \u0627\u0644\u062a\u0644\u0627\u0639\u0628 \u0627\u0644\u062c\u0628\u0631\u064a\u060c \u064a\u0645\u0643\u0646\u0646\u0627 \u0623\u0646 \u0646\u0646\u062a\u0642\u0644 \u0645\u0646 \u0647\u0630\u0627 \u0625\u0644\u0649 \u062d\u0644 \u0622\u062e\u0631 \u0645\u062e\u062a\u0644\u0641 \u062a\u0645\u0627\u0645\u064b\u0627 \u0625\u0630\u0627 \u0644\u0645 \u0646\u0643\u0646 \u0645\u0646\u062a\u0628\u0647\u064a\u0646 \u0644\u0644\u062a\u0641\u0627\u0635\u064a\u0644. \u0639\u0644\u0649 \u0633\u0628\u064a\u0644 \u0627\u0644\u0645\u062b\u0627\u0644\u060c \u0645\u0646 \u0627\u0644\u0648\u0627\u0636\u062d \u0623\u0646:<\/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>\u0648\u0627\u0644\u062c\u0627\u0646\u0628 \u0627\u0644\u0623\u064a\u0645\u0646 \u0645\u0646 \u0647\u0630\u0647 \u0627\u0644\u0645\u0633\u0627\u0648\u0627\u0629 \u0647\u0648 \u0646\u062a\u064a\u062c\u0629 \u0645\u062a\u0633\u0644\u0633\u0644\u0629 \u0647\u0646\u062f\u0633\u064a\u0629:<\/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>\u0644\u0630\u0627 \u0642\u062f \u064a\u0638\u0646 \u0634\u062e\u0635 \u063a\u064a\u0631 \u0645\u062a\u0645\u0631\u0633 \u0641\u064a \u0647\u0630\u0647 \u0627\u0644\u0623\u0645\u0648\u0631 \u0623\u0646 \u0627\u0644\u062f\u0627\u0644\u062a\u064a\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\psi_1<\/span><\/span> \u0648 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\psi_2 = \\sum_{n=0}^{+\\infty} (1-x)^n <\/span><\/span> \u062a\u0642\u062f\u0645\u0627\u0646 \u0646\u0641\u0633 \u0627\u0644\u062d\u0644 \u0644\u0644\u0645\u0639\u0627\u062f\u0644\u0629 \u0627\u0644\u062a\u0641\u0627\u0636\u0644\u064a\u0629 \u0627\u0644\u0645\u0637\u0631\u0648\u062d\u0629 \u0641\u064a \u0627\u0644\u0628\u062f\u0627\u064a\u0629\u060c \u0644\u0623\u0646 \u0646\u062a\u0627\u0626\u062c\u0647\u0645\u0627 \u062a\u0628\u062f\u0648 \u0645\u062a\u0637\u0627\u0628\u0642\u0629\u061b \u0648\u0645\u0639 \u0630\u0644\u0643\u060c \u064a\u063a\u0641\u0644 \u0639\u0646 \u0623\u0646 \u0647\u0630\u0647 \u0627\u0644\u0633\u0644\u0633\u0644\u0629 \u0627\u0644\u0647\u0646\u062f\u0633\u064a\u0629 \u0635\u062d\u064a\u062d\u0629 \u0641\u0642\u0637 \u0639\u0646\u062f\u0645\u0627 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">|1-x| \\lt 1<\/span><\/span>\u060c \u0623\u064a \u0639\u0646\u062f\u0645\u0627 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x\\in]0,2[)<\/span><\/span>. \u0648\u0644\u0643\u0646 \u0647\u0646\u0627\u0643 \u0645\u0627 \u0647\u0648 \u0623\u0643\u062b\u0631\u060c \u0646\u0638\u0631\u064b\u0627 \u0644\u0623\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">]0,2[\\subset]0,+\\infty[<\/span><\/span>\u060c \u0641\u0625\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\psi_1<\/span><\/span> \u062a\u064f\u0639\u062f \u0627\u0645\u062a\u062f\u0627\u062f\u064b\u0627 \u0644\u0640 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\psi_2<\/span><\/span> \u0644\u0623\u0646\u0647 \u0641\u064a \u0627\u0644\u0646\u0637\u0627\u0642 \u0627\u0644\u0630\u064a \u062a\u0643\u0648\u0646 \u0641\u064a\u0647 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\psi_2<\/span><\/span> \u0635\u0627\u0644\u062d\u0629\u060c \u062a\u0643\u0648\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\psi_1<\/span><\/span> \u0643\u0630\u0644\u0643 \u0648\u0623\u064a\u0636\u064b\u0627 \u062e\u0627\u0631\u062c \u0647\u0630\u0627 \u0627\u0644\u0646\u0637\u0627\u0642.<\/p>\n<p><a name=\"SolucionExtendidaYSolucionMaximal\"><\/a><\/p>\n<h4>\u0627\u0644\u062d\u0644 \u0627\u0644\u0645\u0645\u062a\u062f \u0648\u0627\u0644\u062d\u0644 \u0627\u0644\u0623\u0642\u0635\u0649<\/h4>\n<p>\u0644\u0646\u0646\u0638\u0631 \u0641\u064a \u062f\u0627\u0644\u062a\u064a\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\phi_1<\/span><\/span> \u0648 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\phi_2<\/span><\/span> \u0645\u0639\u0631\u0641\u062a\u064a\u0646 \u0639\u0644\u0649 \u0627\u0644\u0641\u062a\u0631\u062a\u064a\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">I_{\\phi_1}<\/span><\/span> \u0648 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">I_{\\phi_2},<\/span><\/span> \u0639\u0644\u0649 \u0627\u0644\u062a\u0648\u0627\u0644\u064a\u060c \u0648\u0627\u0644\u0644\u062a\u064a\u0646 \u0647\u0645\u0627 \u062d\u0644\u0651\u0627\u0646 \u0644\u0645\u0639\u0627\u062f\u0644\u0629 \u062a\u0641\u0627\u0636\u0644\u064a\u0629. \u0625\u0630\u0627 \u0643\u0627\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">I_{\\phi_1}\\subset I_{\\phi_2},<\/span><\/span> \u0641\u0625\u0646\u0646\u0627 \u0646\u0642\u0648\u0644 \u0625\u0646 \u0627\u0644\u062d\u0644 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\phi_2<\/span><\/span> \u064a\u064f\u0645\u062f\u062f \u0627\u0644\u062d\u0644 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\phi_1,<\/span><\/span> \u0623\u0648 \u0623\u0646 \u0627\u0644\u062d\u0644 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\phi_2<\/span><\/span> \u0623\u0643\u062b\u0631 \u0639\u0645\u0648\u0645\u064a\u0629 \u0645\u0646 \u0627\u0644\u062d\u0644 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\phi_1.<\/span><\/span> \u064a\u064f\u0637\u0644\u0642 \u0639\u0644\u0649 \u0627\u0644\u062d\u0644 <span class=\"katex-eq\" data-katex-display=\"false\">\\phi<\/span> \u0627\u0633\u0645 \u00ab\u0627\u0644\u0623\u0642\u0635\u0649\u00bb \u0625\u0630\u0627 \u0644\u0645 \u064a\u0643\u0646 \u0647\u0646\u0627\u0643 \u062d\u0644 \u0622\u062e\u0631 \u064a\u0645\u062f\u062f\u0647 \u0628\u0634\u0643\u0644 \u063a\u064a\u0631 \u062a\u0627\u0641\u0647.<\/p>\n<p><a name=\"SolucionExplicitaYSolucionImplicita\"><\/a><\/p>\n<h4>\u0627\u0644\u062d\u0644 \u0627\u0644\u0635\u0631\u064a\u062d \u0648\u0627\u0644\u062d\u0644 \u0627\u0644\u0636\u0645\u0646\u064a<\/h4>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=zE29azRIKng&#038;t=2649s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #ff0000;\">\u062a\u064f\u0639\u062a\u0628\u0631 \u0627\u0644\u062f\u0627\u0644\u0629<\/span><\/strong><\/a> <span class=\"katex-eq\" data-katex-display=\"false\">\\phi<\/span> \u062d\u0644\u064b\u0627 \u0644\u0644\u0645\u0639\u0627\u062f\u0644\u0629 \u0627\u0644\u062a\u0641\u0627\u0636\u0644\u064a\u0629 \u0627\u0644\u0639\u0627\u062f\u064a\u0629 \u0645\u0646 \u0627\u0644\u0631\u062a\u0628\u0629 <span class=\"katex-eq\" data-katex-display=\"false\">n<\/span> (\u0627\u0644\u0645\u0643\u062a\u0648\u0628\u0629 \u0628\u0627\u0644\u0634\u0643\u0644 \u0627\u0644\u0637\u0628\u064a\u0639\u064a)<\/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> \u0636\u0645\u0646 \u0641\u062a\u0631\u0629 <span class=\"katex-eq\" data-katex-display=\"false\">I<\/span> \u0625\u0630\u0627 \u062a\u062d\u0642\u0642<\/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>\u0645\u0627 \u0631\u0623\u064a\u0646\u0627\u0647 \u0633\u0627\u0628\u0642\u064b\u0627 \u0647\u0648 \u0645\u0627 \u064a\u064f\u0639\u0631\u0641 \u0628\u0640 <strong>\u0627\u0644\u062d\u0644 \u0627\u0644\u0635\u0631\u064a\u062d \u0644\u0644\u0645\u0639\u0627\u062f\u0644\u0629 \u0627\u0644\u062a\u0641\u0627\u0636\u0644\u064a\u0629 \u0639\u0644\u0649 \u0627\u0644\u0641\u062a\u0631\u0629 <span class=\"katex-eq\" data-katex-display=\"false\">I.<\/span><\/strong> \u0648\u0643\u0645\u0627 \u064a\u0634\u064a\u0631 \u0627\u0644\u0627\u0633\u0645\u060c \u0647\u0646\u0627\u0643 \u0623\u064a\u0636\u064b\u0627 \u0637\u0631\u064a\u0642\u0629 \u0636\u0645\u0646\u064a\u0629 \u0644\u062a\u0639\u0631\u064a\u0641 \u0627\u0644\u062d\u0644\u0648\u0644. \u062a\u064f\u0642\u0627\u0644 \u0639\u0644\u0627\u0642\u0629 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\Phi(x,y)=0<\/span><\/span> \u0623\u0646\u0647\u0627 <strong>\u062d\u0644 \u0636\u0645\u0646\u064a \u0644\u0644\u0645\u0639\u0627\u062f\u0644\u0629 \u0627\u0644\u062a\u0641\u0627\u0636\u0644\u064a\u0629 \u0639\u0644\u0649 <span class=\"katex-eq\" data-katex-display=\"false\">I<\/span><\/strong> \u0625\u0630\u0627 \u0643\u0627\u0646\u062a \u062a\u064f\u0639\u0631\u0651\u0641 \u062d\u0644\u064a\u0646 \u0623\u0648 \u0623\u0643\u062b\u0631 \u0628\u0634\u0643\u0644 \u0636\u0645\u0646\u064a \u0641\u064a <span class=\"katex-eq\" data-katex-display=\"false\">I.<\/span>\n<h3>\u0627\u0644\u062e\u0627\u062a\u0645\u0629<\/h3>\n<p>\u0641\u064a \u0647\u0630\u0647 \u0627\u0644\u062d\u0635\u0629 \u0642\u0645\u0646\u0627 \u0628\u062a\u0641\u0643\u064a\u0643 \u0645\u0641\u0647\u0648\u0645 \u0627\u0644\u0645\u0639\u0627\u062f\u0644\u0629 \u0627\u0644\u062a\u0641\u0627\u0636\u0644\u064a\u0629 \u0627\u0644\u0639\u0627\u062f\u064a\u0629 \u0628\u0646\u0638\u0631\u0629 \u062f\u0642\u064a\u0642\u0629 \u0648\u0644\u0643\u0646 \u064a\u0633\u064a\u0631\u0629\u060c \u0648\u0627\u0636\u0639\u064a\u0646 \u0627\u0644\u0623\u0633\u0633 \u0627\u0644\u0634\u0643\u0644\u064a\u0629 \u0627\u0644\u062a\u064a \u062a\u064f\u0645\u0643\u0646\u0646\u0627 \u0644\u064a\u0633 \u0641\u0642\u0637 \u0645\u0646 \u0627\u0644\u062a\u0639\u0631\u0641 \u0639\u0644\u0649 EDO\u060c \u0628\u0644 \u0623\u064a\u0636\u064b\u0627 \u0645\u0646 \u0641\u0647\u0645 \u0627\u0644\u0645\u0646\u0637\u0642 \u0627\u0644\u0643\u0627\u0645\u0646 \u0648\u0631\u0627\u0621 \u062d\u0644\u0648\u0644\u0647\u0627. \u0648\u0628\u0641\u0636\u0644 \u0645\u0628\u0631\u0647\u0646\u0629 \u0627\u0644\u062f\u0627\u0644\u0629 \u0627\u0644\u0636\u0645\u0646\u064a\u0629\u060c \u0643\u0627\u0646 \u0645\u0646 \u0627\u0644\u0645\u0645\u0643\u0646 \u062a\u0628\u0631\u064a\u0631 \u0627\u0644\u0627\u0646\u062a\u0642\u0627\u0644 \u0628\u0648\u0636\u0648\u062d \u0628\u064a\u0646 \u0634\u0643\u0644\u0647\u0627 \u0627\u0644\u0639\u0627\u0645 \u0648\u0634\u0643\u0644\u0647\u0627 \u0627\u0644\u0637\u0628\u064a\u0639\u064a\u060c \u0645\u0645\u0627 \u064a\u064f\u062a\u0631\u062c\u0645 \u0625\u0644\u0649 \u0642\u062f\u0631\u0629 \u062a\u0642\u0646\u064a\u0629 \u0623\u0633\u0627\u0633\u064a\u0629 \u0644\u0645\u0639\u0627\u0644\u062c\u0629 \u0627\u0644\u0645\u0633\u0627\u0626\u0644 \u0627\u0644\u062a\u0637\u0628\u064a\u0642\u064a\u0629.<\/p>\n<p>\u0628\u0627\u0644\u0625\u0636\u0627\u0641\u0629 \u0625\u0644\u0649 \u0630\u0644\u0643\u060c \u0645\u064a\u0651\u0632\u0646\u0627 \u0628\u062f\u0642\u0629 \u0628\u064a\u0646 \u0627\u0644\u0637\u0631\u0642 \u0627\u0644\u0645\u062e\u062a\u0644\u0641\u0629 \u0627\u0644\u062a\u064a \u064a\u0645\u0643\u0646 \u0623\u0646 \u064a\u064f\u0641\u0647\u0645 \u0628\u0647\u0627 \u0627\u0644\u062d\u0644: \u0643\u062d\u0644 \u0635\u0631\u064a\u062d \u0623\u0648 \u0636\u0645\u0646\u064a\u060c \u0645\u0645\u062a\u062f \u0623\u0648 \u0623\u0642\u0635\u0649\u060c \u0648\u0623\u0628\u0631\u0632\u0646\u0627 \u0623\u0647\u0645\u064a\u0629 \u2014\u063a\u0627\u0644\u0628\u064b\u0627 \u0645\u0627 \u064a\u064f\u0633\u062a\u0647\u0627\u0646 \u0628\u0647\u0627\u2014 \u0641\u064a \u0627\u0644\u062a\u0635\u0631\u064a\u062d \u0628\u0634\u0643\u0644 \u0645\u0646\u0627\u0633\u0628 \u0639\u0646 \u0645\u062c\u0627\u0644\u0647. \u0647\u0630\u0647 \u0627\u0644\u062a\u0645\u064a\u064a\u0632\u0627\u062a \u0644\u064a\u0633\u062a \u0641\u0642\u0637 \u0634\u0643\u0644\u064a\u0629: \u0625\u0646\u0647\u0627 \u0639\u0645\u0644\u064a\u0629. \u0625\u0646 \u062a\u062c\u0627\u0647\u0644\u0647\u0627 \u0642\u062f \u064a\u0642\u0648\u062f\u0646\u0627\u060c \u0643\u0645\u0627 \u0631\u0623\u064a\u0646\u0627\u060c \u0625\u0644\u0649 \u0623\u062e\u0637\u0627\u0621 \u0645\u0641\u0627\u0647\u064a\u0645\u064a\u0629 \u062c\u0633\u064a\u0645\u0629 \u0639\u0646\u062f \u062a\u0641\u0633\u064a\u0631 \u0627\u0644\u0646\u062a\u0627\u0626\u062c \u0627\u0644\u0645\u062d\u0635\u0651\u0644\u0629.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u0645\u0627 \u0647\u064a \u0627\u0644\u0645\u0639\u0627\u062f\u0644\u0629 \u0627\u0644\u062a\u0641\u0627\u0636\u0644\u064a\u0629 \u0627\u0644\u0639\u0627\u062f\u064a\u0629 (EDO)\u061f \u0627\u0644\u0645\u0644\u062e\u0635:\u0641\u064a \u0647\u0630\u0647 \u0627\u0644\u062d\u0635\u0629\u060c \u064a\u062a\u0645 \u0627\u0633\u062a\u0643\u0634\u0627\u0641 \u0627\u0644\u0645\u0639\u0627\u062f\u0644\u0627\u062a \u0627\u0644\u062a\u0641\u0627\u0636\u0644\u064a\u0629 \u0627\u0644\u0639\u0627\u062f\u064a\u0629 (EDO) \u0645\u0646 \u0627\u0644\u0631\u062a\u0628\u0629 k\u060c \u0628\u062f\u0621\u064b\u0627 \u0645\u0646 \u062a\u0639\u0631\u064a\u0641\u0647\u0627 \u0648\u062a\u0645\u062b\u064a\u0644\u0647\u0627 \u0628\u0627\u0644\u0634\u0643\u0644\u064a\u0646 \u0627\u0644\u0639\u0627\u062f\u064a \u0648\u0627\u0644\u0639\u0627\u0645. \u0645\u0646 \u062e\u0644\u0627\u0644 \u0645\u0641\u0627\u0647\u064a\u0645 \u0645\u062b\u0644 \u0645\u0635\u0641\u0648\u0641\u0629 \u064a\u0639\u0642\u0648\u0628\u064a \u0648\u0645\u0628\u0631\u0647\u0646\u0629 \u0627\u0644\u062f\u0627\u0644\u0629 \u0627\u0644\u0636\u0645\u0646\u064a\u0629\u060c \u064a\u062a\u0645 \u0648\u0636\u0639 \u0627\u0644\u0623\u0633\u0627\u0633 \u0644\u0641\u0647\u0645 \u062d\u0644\u0648\u0644 \u0647\u0630\u0647 \u0627\u0644\u0645\u0639\u0627\u062f\u0644\u0627\u062a \u0648\u0627\u0644\u062e\u0635\u0627\u0626\u0635 \u0627\u0644\u0645\u0631\u062a\u0628\u0637\u0629 \u0628\u0647\u0627\u060c \u0645\u062b\u0644 \u0645\u062c\u0627\u0644 \u0627\u0644\u062a\u0639\u0631\u064a\u0641 \u0648\u0627\u0644\u062d\u0644\u0648\u0644 \u0627\u0644\u0635\u0631\u064a\u062d\u0629 \u0648\u0627\u0644\u0636\u0645\u0646\u064a\u0629. \u0623\u0647\u0640\u0640\u062f\u0627\u0641 \u0627\u0644\u062a\u0639\u0644\u0640\u0640\u0645 \u0639\u0646\u062f \u0646\u0647\u0627\u064a\u0629 \u0647\u0630\u0647 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