{"id":34413,"date":"2022-04-28T13:00:49","date_gmt":"2022-04-28T13:00:49","guid":{"rendered":"https:\/\/toposuranos.com\/material\/?p=34413"},"modified":"2025-09-08T00:16:03","modified_gmt":"2025-09-08T00:16:03","slug":"quid-est-aequatio-differentiae-ordinaria-ado","status":"publish","type":"post","link":"https:\/\/toposuranos.com\/material\/la\/quid-est-aequatio-differentiae-ordinaria-ado\/","title":{"rendered":"Quid est Aequatio Differentiae Ordinaria (ADO)?"},"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>Quid est Aequatio Differentialis Ordinaria (ADO)?<\/h1>\n<p style=\"text-align:center;\" dir=\"ltr\"><em><strong>Summarium:<\/strong><\/br>In hac lectione explorantur Aequationes Differentiales Ordinariae (ADO) ordinis k, incipiendo a definitione earum atque a repraesentatione forma normali et generali. Per conceptus sicut matrix Jacobiana et Theorema Functionis Implicitae, fundamenta ponuntur ad intellegendum solutiones harum aequationum atque proprietates associatas, ut dominium definitionis et solutiones explicitae ac implicitae.<\/em><\/p>\n<p style=\"text-align:center\"><strong>PROPOSITA DISCENDI<\/strong><\/p>\n<p>In fine huius lectionis discipulus poterit:<\/p>\n<ol>\n<li><strong>Meminisse<\/strong> definitionem et notas fundamentales Aequationis Differentialis Ordinariae (ADO).<\/li>\n<li><strong>Explicare<\/strong> relationem inter ADO et solutiones eius posse haberi.<\/li>\n<\/ol>\n<p style=\"text-align:center;\" dir=\"ltr\"><strong>INDEX<\/strong><br \/>\n<a href=\"#LaEcuacionDiferencialOrdinariaDeOrdenK\"><strong>Aequatio Differentiae Ordinaria (ADO) Ordinis k<\/strong><\/a><br \/>\n<a href=\"#TeoremaDeLaFuncionImplicita\">Theorema Functionis Implicitae<\/a><br \/>\n<a href=\"#LaSolucionDeUnaEcuacionDiferencialOrdinaria\"><strong>Solutio Aequationis Differentialis Ordinariae<\/strong><\/a><br \/>\n<a href=\"#CuidadoConElDominioDeDefinicionDeLasSoluciones\">Cavendum de dominio definitionis solutionum<\/a><br \/>\n<a href=\"#SolucionExtendidaYSolucionMaximal\">Solutio extensa et solutio maxima<\/a><br \/>\n<a href=\"#SolucionExplicitaYSolucionImplicita\">Solutio explicita et solutio implicita<\/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>His quae hucusque visa sunt, satis clara idea nobis est quid sit aequatio differentialis et multiplices applicationes quas habere potest. Nunc consistemus ut aliquas definitiones et proprietates studeamus, proposito fundamentum commune firmum statuere ad hoc studium continuandum.<\/p>\n<p><a name=\"LaEcuacionDiferencialOrdinariaDeOrdenK\"><\/a><\/p>\n<h3>ADO Ordinis 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;\">Aequatio Differentiae Ordinaria (ADO)<\/span><\/strong><\/a> est aequatio in qua implicantur variabilis independens <span class=\"katex-eq\" data-katex-display=\"false\">x<\/span>, functio <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">y(x)<\/span><\/span>, et nonnullae eius derivatae ordinariae. Derivatae ordinariae primi ordinis <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">y(x)<\/span><\/span> notantur symbolis ut <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\frac{dy(x)}{dx}<\/span><\/span> vel <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">y&#039;(x)<\/span><\/span>, secundi ordinis ut <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\frac{d^2y(x)}{dx^2}<\/span><\/span> vel <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">y&#039;&#039;(x)<\/span><\/span>, et in genere, ordinis <span class=\"katex-eq\" data-katex-display=\"false\">n<\/span>, ut <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\frac{d^ny(x)}{dx^n}<\/span><\/span> vel <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">y^{(n)}(x)<\/span><\/span>. Supremum valorum <span class=\"katex-eq\" data-katex-display=\"false\">k<\/span> talium ut <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">y^{(k)}(x)<\/span><\/span> appareat in aequatione est quod vocamus <strong>Ordo Aequationis<\/strong>. Itaque, <strong>Forma Generalis ADO ordinis <span class=\"katex-eq\" data-katex-display=\"false\">k<\/span><\/strong> est:<\/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>Dicitur ADO ordinis <span class=\"katex-eq\" data-katex-display=\"false\">k<\/span> esse in <strong>forma normali<\/strong> si exprimitur solvendo <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">y^{(k)}(x)<\/span><\/span> ex aequatione superiore, id est:<\/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>In genere, functio <span class=\"katex-eq\" data-katex-display=\"false\">y<\/span> est functio <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R} \\longrightarrow \\mathbb{R}^n,<\/span><\/span> ita ut haec et omnes eius derivatae in aliquo puncto <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x\\in\\mathbb{R}<\/span><\/span> evaluatae sint vectores in <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n<\/span><\/span>. Hoc considerato, constat quod, cum functio <span class=\"katex-eq\" data-katex-display=\"false\">F<\/span> quae describit ADO ordinis <span class=\"katex-eq\" data-katex-display=\"false\">k<\/span> habeat <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">1+(k+1)<\/span><\/span> variabiles, habetur <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\text{Dom}(F)\\subset \\mathbb{R}^{1+n(k+1)}<\/span><\/span> et <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\text{Rec}(F)\\subset \\mathbb{R}<\/span><\/span>; et similiter, <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\text{Dom}(f) = \\mathbb{R}^{1+nk}<\/span><\/span> et <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\text{Rec}(f)\\subset \\mathbb{R}^n<\/span><\/span>.<\/p>\n<p>Transitus ab expressione Generali ADO ordinis <span class=\"katex-eq\" data-katex-display=\"false\">k<\/span> ad Formam Normalem fieri potest gratia <strong>Theorematis Functionis Implicitae.<\/strong><\/p>\n<p><a name=\"TeoremaDeLaFuncionImplicita\"><\/a><\/p>\n<h4>Theorema Functionis Implicitae<\/h4>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=zE29azRIKng&#038;t=887s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #ff0000;\">Sit <span class=\"katex-eq\" data-katex-display=\"false\">F<\/span> functio classis <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathcal{C}^1<\/span><\/span> super coniunctum apertum <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">U \\subset \\mathbb{R}^n<\/span><\/span><\/span><\/strong><\/a> cum valoribus realibus. Et sit <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(a_1,\\cdots, a_n) \\in U<\/span><\/span> tale ut <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F(a_1,\\cdots, a_n) = 0<\/span><\/span> et<\/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>Tunc exstat vicinitas <span class=\"katex-eq\" data-katex-display=\"false\">V<\/span> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(a_1, \\cdots, a_{n-1}) \\in \\mathbb{R}^{n-1}<\/span><\/span> et functio <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\varphi:V \\longrightarrow \\mathbb{R}<\/span><\/span> talis ut:<\/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> est differentiabilis et\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>Demonstratio Theorematis Functionis Implicitae<\/h4>\n<h5>Evolutio ex matrice Jacobiana<\/h5>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=zE29azRIKng&#038;t=1101s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #ff0000;\">Sit <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> Si computamus eius matricem Jacobianam, quae infra monstratur:<\/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>videbimus eius determinantem esse diversum a nullo in <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(a_1,\\cdots, a_n)<\/span><\/span>, prorsus quia, ut initio statutum est, <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> Ex hoc possumus dicere <span class=\"katex-eq\" data-katex-display=\"false\">\\psi<\/span> habere inversam super coniunctum apertum <span class=\"katex-eq\" data-katex-display=\"false\">W<\/span> quod continet <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(a_1,\\cdots, a_n).<\/span><\/span><\/p>\n<h5>Evolutio Solutionis<\/h5>\n<p>Nunc consideremus coniunctum<\/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>Ex hoc possumus definire aliud coniunctum<\/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>Coniunctum <span class=\"katex-eq\" data-katex-display=\"false\">V<\/span> est, proinde, apertum quod continet <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>Praeterea, cum <span class=\"katex-eq\" data-katex-display=\"false\">\\psi<\/span> habeat inversam (in <span class=\"katex-eq\" data-katex-display=\"false\">W<\/span>), exstat unicum <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(y_1,\\cdots,y_n)\\in W<\/span><\/span> tale ut <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> Hoc significat:<\/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>Ita possumus definire <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\varphi(x_1,\\cdots,x_{n-1}) = y_n<\/span><\/span>, ita ut:<\/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>et<\/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>Ex hoc habemus quod <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\varphi(V)\\ni a_n,<\/span><\/span> et proinde <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">V\\times\\varphi(V) \\subset U,<\/span><\/span> et insuper:<\/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>Differentiabilitas<\/h5>\n<p>Et denique, differentiabilitas <span class=\"katex-eq\" data-katex-display=\"false\">\\psi<\/span> ducit ad differentiabilitatem <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\psi^{-1}<\/span><\/span>, quae vicissim ducit ad differentiabilitatem <span class=\"katex-eq\" data-katex-display=\"false\">\\varphi<\/span> super <span class=\"katex-eq\" data-katex-display=\"false\">V<\/span>. Hoc considerato, possumus definire functionem <span class=\"katex-eq\" data-katex-display=\"false\">g<\/span> per relationem:<\/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>Et postea, utens regula catenae, habetur:<\/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>ubi <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">i=1,\\cdots, n-1.<\/span><\/span> Ex hac ultima aequatione obtinetur:<\/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>Et hoc cum sit, concluditur omnia quae demonstrari volebantur \u25a0<\/p>\n<p><a name=\"LaSolucionDeUnaEcuacionDiferencialOrdinaria\"><\/a><\/p>\n<h3>Solutio aequationis differentialis ordinariae<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=zE29azRIKng&#038;t=2249s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #ff0000;\">Consideremus ADO expressam in forma normali<\/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>Tunc functio <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\varphi : I_\\phi \\longmapsto \\mathbb{R}^n,<\/span><\/span> ubi <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">I_\\phi<\/span><\/span> est intervallum <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R},<\/span><\/span> dicitur esse <strong>solutio ADO<\/strong> si:<\/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>Cavendum de dominio definitionis solutionum<\/h4>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=zE29azRIKng&#038;t=2387s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #ff0000;\">Hoc loco, necesse est emphasin ponere<\/span><\/strong><\/a> de momento declarandi expresse dominium solutionis aequationis differentialis. Exempli gratia, dominium functionis <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\phi<\/span><\/span> de qua locuti sumus in praecedente paragrapho est intervallum <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">I_\\phi.<\/span><\/span> Hoc est grave quia error communis in operando cum aequationibus differentialibus provenit ex eo quod duo solutiones <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\phi_1<\/span><\/span> et <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\phi_2<\/span><\/span> aequales putantur solum quia <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> quamvis <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">I_{\\phi_1}\\neq I_{\\phi_2}.<\/span><\/span> Ad explicandum hoc punctum, examinemus aequationem differentialem:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">y^\\prime = -y^2.<\/span>\n<p>Una solutio possibilis huius ADO est functio <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\psi_1 : ]0,+\\infty[ \\longrightarrow \\mathbb{R}^+\\setminus\\{0\\}<\/span><\/span> definita per <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\psi_1(x)=1\/x,<\/span><\/span> quia <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\psi_1^{\\prime} = -1\/x^2 = -\\psi_1^2<\/span><\/span> pro quolibet <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x\\in]0,+\\infty[.<\/span><\/span> Sed aliquantulo ludo algebraico, transire possumus ex hac ad aliam solutionem omnino diversam nisi attendamus ad singula. Exempli gratia, manifestum est quod:<\/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>et latus dextrum huius aequalitatis est effectus seriei geometricae:<\/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>Ita oculus parum exercitatus in his artibus arcanis auderet putare functiones <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\psi_1<\/span><\/span><br \/>\n et <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\psi_2 = \\sum_{n=0}^{+\\infty} (1-x)^n <\/span><\/span> nobis eandem solutionem praebere aequationi differentiali initio propositae, quia revera in resultatis conveniunt; attamen praeterierit quod haec series geometrica tantum valet cum <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">|1-x| \\lt 1<\/span><\/span>, id est, cum <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x\\in]0,2[)<\/span><\/span>. Sed est plus: cum <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">]0,2[\\subset]0,+\\infty[<\/span><\/span>, etiam habetur quod <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\psi_1<\/span><\/span> extendit ad <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\psi_2<\/span><\/span> quia ibi ubi <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\psi_2<\/span><\/span> valet, <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\psi_1<\/span><\/span> valet atque etiam ultra.<\/p>\n<p><a name=\"SolucionExtendidaYSolucionMaximal\"><\/a><\/p>\n<h4>Solutio extensa et solutio maxima<\/h4>\n<p>Consideremus duas functiones <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\phi_1<\/span><\/span> et <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\phi_2<\/span><\/span> definitas super intervalla <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">I_{\\phi_1}<\/span><\/span> et <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">I_{\\phi_2},<\/span><\/span> respective, quae sunt solutiones aequationis differentialis. Si <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">I_{\\phi_1}\\subset I_{\\phi_2},<\/span><\/span> dicitur tunc solutionem <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\phi_2<\/span><\/span> extendere solutionem <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\phi_1,<\/span><\/span> aut solutionem <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\phi_2<\/span><\/span> esse generalior quam solutio <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\phi_1.<\/span><\/span> Solutio <span class=\"katex-eq\" data-katex-display=\"false\">\\phi<\/span> appellatur \u00abmaxima\u00bb si nulla alia solutio exstat quae eam extendat non trivialiter.<\/p>\n<p><a name=\"SolucionExplicitaYSolucionImplicita\"><\/a><\/p>\n<h4>Solutio explicita et solutio implicita<\/h4>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=zE29azRIKng&#038;t=2649s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #ff0000;\">Functio quaedam<\/span><\/strong><\/a> <span class=\"katex-eq\" data-katex-display=\"false\">\\phi<\/span> censetur solutio ADO ordinis <span class=\"katex-eq\" data-katex-display=\"false\">n<\/span> (scriptae in forma normali)<\/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> intra intervallum <span class=\"katex-eq\" data-katex-display=\"false\">I<\/span> si<\/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>Quod iam antea per aliquot paragraphos retractavimus est quod cognoscitur ut <strong>Solutio Explicita Aequationis Differentiae in intervallo <span class=\"katex-eq\" data-katex-display=\"false\">I.<\/span><\/strong> Sicut nomen indicat, etiam exstat forma implicita ad definire solutiones. Dicitur relatio <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\Phi(x,y)=0<\/span><\/span> esse <strong>Solutio Implicita Aequationis Differentiae in <span class=\"katex-eq\" data-katex-display=\"false\">I<\/span><\/strong> si definit duas vel plures solutiones implicitas in <span class=\"katex-eq\" data-katex-display=\"false\">I.<\/span>\n<h3>Conclusio<\/h3>\n<p>In hac lectione notionem aequationis differentialis ordinariae rigida sed accessibili perspectiva decomponebamus, fundamenta formalia statuentes quae nobis permittunt non solum agnoscere ADO, sed etiam intelligere rationem quae post solutiones eius latet. Per Theorema Functionis Implicitae, possibile fuit clare iustificare transitionem inter formam generalem et formam normalem, quod vertitur in facultatem technicam crucialem ad tractandos problemas concretos.<\/p>\n<p>Praeterea, distincte discrevimus varias rationes quibus solutio comprehendi potest: ut solutio explicita vel implicita, extensa vel maxima, atque notavimus momentum \u2014saepe subestimatum\u2014 recte declarandi eius dominium. Hae distinctiones non sunt solum formales: sunt operativae. Negligere eas nos ducere potest, ut vidimus, ad errores graves conceptuales in interpretandis resultatis obtentis.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Quid est Aequatio Differentialis Ordinaria (ADO)? Summarium:In hac lectione explorantur Aequationes Differentiales Ordinariae (ADO) ordinis k, incipiendo a definitione earum atque a repraesentatione forma normali et generali. Per conceptus sicut matrix Jacobiana et Theorema Functionis Implicitae, fundamenta ponuntur ad intellegendum solutiones harum aequationum atque proprietates associatas, ut dominium definitionis et solutiones explicitae ac implicitae. PROPOSITA [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":32838,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":2,"footnotes":""},"categories":[1346,1298,1186],"tags":[],"class_list":["post-34413","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-aequationes-differentiae-ordinariae","category-mathematica","category-sin-categoria-la"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v26.7 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Quid est Aequatio Differentiae Ordinaria (ADO)? - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"Disce quid sit Aequatio Differentiae Ordinaria et disce eius fundamenta clare, rigide atque exemplis concretis percipere.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/toposuranos.com\/material\/la\/quid-est-aequatio-differentiae-ordinaria-ado\/\" \/>\n<meta property=\"og:locale\" content=\"es_ES\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Quid est Aequatio Differentiae Ordinaria (ADO)?\" \/>\n<meta property=\"og:description\" content=\"Disce quid sit Aequatio Differentiae Ordinaria et disce eius fundamenta clare, rigide atque exemplis concretis percipere.\" \/>\n<meta property=\"og:url\" content=\"https:\/\/toposuranos.com\/material\/la\/quid-est-aequatio-differentiae-ordinaria-ado\/\" \/>\n<meta property=\"og:site_name\" content=\"toposuranos.com\/material\" \/>\n<meta property=\"article:publisher\" content=\"https:\/\/www.facebook.com\/groups\/toposuranos\" \/>\n<meta property=\"article:published_time\" content=\"2022-04-28T13:00:49+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2025-09-08T00:16:03+00:00\" \/>\n<meta property=\"og:image\" content=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2022\/04\/edo-1024x381.jpg\" \/>\n<meta name=\"author\" content=\"giorgio.reveco\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:title\" content=\"Quid est Aequatio Differentiae Ordinaria (ADO)?\" \/>\n<meta name=\"twitter:description\" content=\"Disce quid sit Aequatio Differentiae Ordinaria et disce eius fundamenta clare, rigide atque exemplis concretis percipere.\" \/>\n<meta name=\"twitter:image\" content=\"https:\/\/toposuranos.com\/material\/wp-content\/uploads\/2022\/04\/edo.jpg\" \/>\n<meta name=\"twitter:creator\" content=\"@topuranos\" \/>\n<meta name=\"twitter:site\" content=\"@topuranos\" \/>\n<meta name=\"twitter:label1\" content=\"Escrito por\" \/>\n\t<meta name=\"twitter:data1\" content=\"giorgio.reveco\" \/>\n\t<meta name=\"twitter:label2\" content=\"Tiempo de lectura\" \/>\n\t<meta name=\"twitter:data2\" content=\"1 minuto\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\/\/toposuranos.com\/material\/la\/quid-est-aequatio-differentiae-ordinaria-ado\/#article\",\"isPartOf\":{\"@id\":\"https:\/\/toposuranos.com\/material\/la\/quid-est-aequatio-differentiae-ordinaria-ado\/\"},\"author\":{\"name\":\"giorgio.reveco\",\"@id\":\"https:\/\/toposuranos.com\/material\/#\/schema\/person\/e15164361c3f9a2a02cf6c234cf7fdc1\"},\"headline\":\"Quid est Aequatio Differentiae Ordinaria (ADO)?\",\"datePublished\":\"2022-04-28T13:00:49+00:00\",\"dateModified\":\"2025-09-08T00:16:03+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\/\/toposuranos.com\/material\/la\/quid-est-aequatio-differentiae-ordinaria-ado\/\"},\"wordCount\":1770,\"commentCount\":0,\"publisher\":{\"@id\":\"https:\/\/toposuranos.com\/material\/#organization\"},\"image\":{\"@id\":\"https:\/\/toposuranos.com\/material\/la\/quid-est-aequatio-differentiae-ordinaria-ado\/#primaryimage\"},\"thumbnailUrl\":\"https:\/\/toposuranos.com\/material\/wp-content\/uploads\/2022\/04\/edo.jpg\",\"articleSection\":[\"Aequationes Differentiae Ordinariae\",\"Mathematica\"],\"inLanguage\":\"es\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"https:\/\/toposuranos.com\/material\/la\/quid-est-aequatio-differentiae-ordinaria-ado\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"https:\/\/toposuranos.com\/material\/la\/quid-est-aequatio-differentiae-ordinaria-ado\/\",\"url\":\"https:\/\/toposuranos.com\/material\/la\/quid-est-aequatio-differentiae-ordinaria-ado\/\",\"name\":\"Quid est Aequatio Differentiae Ordinaria (ADO)? 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