. "Linear differential equation"@en . . . . . "Ecuaci\u00F3n diferencial lineal"@es . "Equaci\u00F3 diferencial lineal"@ca . "\u0412 \u043C\u0430\u0442\u0435\u043C\u0430\u0442\u0438\u043A\u0435 \u043B\u0438\u043D\u0435\u0439\u043D\u043E\u0435 \u0434\u0438\u0444\u0444\u0435\u0440\u0435\u043D\u0446\u0438\u0430\u043B\u044C\u043D\u043E\u0435 \u0443\u0440\u0430\u0432\u043D\u0435\u043D\u0438\u0435 \u0438\u043C\u0435\u0435\u0442 \u0432\u0438\u0434 \u0433\u0434\u0435 \u0434\u0438\u0444\u0444\u0435\u0440\u0435\u043D\u0446\u0438\u0430\u043B\u044C\u043D\u044B\u0439 \u043E\u043F\u0435\u0440\u0430\u0442\u043E\u0440 L \u043B\u0438\u043D\u0435\u0435\u043D, y \u2014 \u0438\u0437\u0432\u0435\u0441\u0442\u043D\u0430\u044F \u0444\u0443\u043D\u043A\u0446\u0438\u044F , \u0430 \u043F\u0440\u0430\u0432\u0430\u044F \u0447\u0430\u0441\u0442\u044C \u2014 \u0444\u0443\u043D\u043A\u0446\u0438\u044F \u043E\u0442 \u0442\u043E\u0439 \u0436\u0435 \u043F\u0435\u0440\u0435\u043C\u0435\u043D\u043D\u043E\u0439, \u0447\u0442\u043E \u0438 y. \u041B\u0438\u043D\u0435\u0439\u043D\u044B\u0439 \u043E\u043F\u0435\u0440\u0430\u0442\u043E\u0440 L \u043C\u043E\u0436\u043D\u043E \u0440\u0430\u0441\u0441\u043C\u0430\u0442\u0440\u0438\u0432\u0430\u0442\u044C \u0432 \u0444\u043E\u0440\u043C\u0435 \u041F\u0440\u0438 \u044D\u0442\u043E\u043C, \u0435\u0441\u043B\u0438 , \u0442\u043E \u0442\u0430\u043A\u043E\u0435 \u0443\u0440\u0430\u0432\u043D\u0435\u043D\u0438\u0435 \u043D\u0430\u0437\u044B\u0432\u0430\u0435\u0442\u0441\u044F \u043B\u0438\u043D\u0435\u0439\u043D\u044B\u043C \u043E\u0434\u043D\u043E\u0440\u043E\u0434\u043D\u044B\u043C \u0443\u0440\u0430\u0432\u043D\u0435\u043D\u0438\u0435\u043C, \u0438\u043D\u0430\u0447\u0435 \u2014 \u043B\u0438\u043D\u0435\u0439\u043D\u044B\u043C \u043D\u0435\u043E\u0434\u043D\u043E\u0440\u043E\u0434\u043D\u044B\u043C \u0443\u0440\u0430\u0432\u043D\u0435\u043D\u0438\u0435\u043C."@ru . . "Line\u00E1rn\u00ED diferenci\u00E1ln\u00ED rovnice"@cs . . . . "\u7EBF\u6027\u5FAE\u5206\u65B9\u7A0B"@zh . . . . . . "Lineaire differentiaalvergelijking van eerste orde"@nl . . . . "Lineare gew\u00F6hnliche Differentialgleichung"@de . . "\u0645\u0639\u0627\u062F\u0644\u0629 \u062A\u0641\u0627\u0636\u0644\u064A\u0629 \u062E\u0637\u064A\u0629"@ar . . "1106453723"^^ . . . . . . . "\u7EBF\u6027\u5FAE\u5206\u65B9\u7A0B\uFF08\u82F1\u8A9E\uFF1ALinear differential equation\uFF09\u662F\u6570\u5B66\u4E2D\u5E38\u89C1\u7684\u4E00\u7C7B\u5FAE\u5206\u65B9\u7A0B\u3002\u6307\u4EE5\u4E0B\u5F62\u5F0F\u7684\u5FAE\u5206\u65B9\u7A0B\uFF1A \u5176\u4E2D\u65B9\u7A0B\u5DE6\u4FA7\u7684\u5FAE\u5206\u7B97\u5B50\u662F\u7EBF\u6027\u7B97\u5B50\uFF0Cy\u662F\u8981\u89E3\u7684\u672A\u77E5\u51FD\u6570\uFF0C\u65B9\u7A0B\u7684\u53F3\u4FA7\u662F\u4E00\u4E2A\u5DF2\u77E5\u51FD\u6570\u3002\u5982\u679Cf(x) = 0\uFF0C\u90A3\u4E48\u65B9\u7A0B(*)\u7684\u89E3\u7684\u7EBF\u6027\u7EC4\u5408\u4ECD\u7136\u662F\u89E3\uFF0C\u6240\u6709\u7684\u89E3\u6784\u6210\u4E00\u4E2A\u5411\u91CF\u7A7A\u95F4\uFF0C\u79F0\u4E3A\u89E3\u7A7A\u95F4\u3002\u8FD9\u6837\u7684\u65B9\u7A0B\u79F0\u4E3A\u9F50\u6B21\u7EBF\u6027\u5FAE\u5206\u65B9\u7A0B\u3002\u5F53f\u4E0D\u662F\u96F6\u51FD\u6570\u65F6\uFF0C\u6240\u6709\u7684\u89E3\u6784\u6210\u4E00\u4E2A\u4EFF\u5C04\u7A7A\u95F4\uFF0C\u7531\u5BF9\u5E94\u7684\u9F50\u6B21\u65B9\u7A0B\u7684\u89E3\u7A7A\u95F4\u52A0\u4E0A\u4E00\u4E2A\u7279\u89E3\u5F97\u5230\u3002\u8FD9\u6837\u7684\u65B9\u7A0B\u79F0\u4E3A\u975E\u9F50\u6B21\u7EBF\u6027\u5FAE\u5206\u65B9\u7A0B\u3002\u7EBF\u6027\u5FAE\u5206\u65B9\u7A0B\u53EF\u4EE5\u662F\u5E38\u5FAE\u5206\u65B9\u7A0B\uFF0C\u4E5F\u53EF\u4EE5\u662F\u504F\u5FAE\u5206\u65B9\u7A0B\u3002"@zh . . "En matem\u00E0tiques, les equacions diferencials lineals s\u00F3n equacions diferencials que tenen solucions que poden sumar-se per obtenir altres solucions. Poden ser ordin\u00E0ries (EDOs) o parcials (EDPs). Les solucions d'equacions linears formen un espai vectorial (a difer\u00E8ncia de les ). Una equaci\u00F3 diferencial lineal \u00E9s una equaci\u00F3 diferencial que t\u00E9 i, fins i tot m\u00E9s precisament L'operador lineal L es pot considerar de la forma. on D \u00E9s l'operador diferencial d/dt (\u00E9s a dir Dy = y, D \u00B2y = y\"... ), i An s\u00F3n funcions donades. Tal equaci\u00F3 es diu que t\u00E9 ordre n, l'\u00EDndex de la derivada m\u00E9s alt de y."@ca . . . . . . . . . "\u0412 \u043C\u0430\u0442\u0435\u043C\u0430\u0442\u0438\u043A\u0435 \u043B\u0438\u043D\u0435\u0439\u043D\u043E\u0435 \u0434\u0438\u0444\u0444\u0435\u0440\u0435\u043D\u0446\u0438\u0430\u043B\u044C\u043D\u043E\u0435 \u0443\u0440\u0430\u0432\u043D\u0435\u043D\u0438\u0435 \u0438\u043C\u0435\u0435\u0442 \u0432\u0438\u0434 \u0433\u0434\u0435 \u0434\u0438\u0444\u0444\u0435\u0440\u0435\u043D\u0446\u0438\u0430\u043B\u044C\u043D\u044B\u0439 \u043E\u043F\u0435\u0440\u0430\u0442\u043E\u0440 L \u043B\u0438\u043D\u0435\u0435\u043D, y \u2014 \u0438\u0437\u0432\u0435\u0441\u0442\u043D\u0430\u044F \u0444\u0443\u043D\u043A\u0446\u0438\u044F , \u0430 \u043F\u0440\u0430\u0432\u0430\u044F \u0447\u0430\u0441\u0442\u044C \u2014 \u0444\u0443\u043D\u043A\u0446\u0438\u044F \u043E\u0442 \u0442\u043E\u0439 \u0436\u0435 \u043F\u0435\u0440\u0435\u043C\u0435\u043D\u043D\u043E\u0439, \u0447\u0442\u043E \u0438 y. \u041B\u0438\u043D\u0435\u0439\u043D\u044B\u0439 \u043E\u043F\u0435\u0440\u0430\u0442\u043E\u0440 L \u043C\u043E\u0436\u043D\u043E \u0440\u0430\u0441\u0441\u043C\u0430\u0442\u0440\u0438\u0432\u0430\u0442\u044C \u0432 \u0444\u043E\u0440\u043C\u0435 \u041F\u0440\u0438 \u044D\u0442\u043E\u043C, \u0435\u0441\u043B\u0438 , \u0442\u043E \u0442\u0430\u043A\u043E\u0435 \u0443\u0440\u0430\u0432\u043D\u0435\u043D\u0438\u0435 \u043D\u0430\u0437\u044B\u0432\u0430\u0435\u0442\u0441\u044F \u043B\u0438\u043D\u0435\u0439\u043D\u044B\u043C \u043E\u0434\u043D\u043E\u0440\u043E\u0434\u043D\u044B\u043C \u0443\u0440\u0430\u0432\u043D\u0435\u043D\u0438\u0435\u043C, \u0438\u043D\u0430\u0447\u0435 \u2014 \u043B\u0438\u043D\u0435\u0439\u043D\u044B\u043C \u043D\u0435\u043E\u0434\u043D\u043E\u0440\u043E\u0434\u043D\u044B\u043C \u0443\u0440\u0430\u0432\u043D\u0435\u043D\u0438\u0435\u043C."@ru . . "\u7DDA\u578B\u5FAE\u5206\u65B9\u7A0B\u5F0F"@ja . . . . "Een lineaire differentiaalvergelijking van eerste orde is een speciaal geval van een lineaire differentiaalvergelijking, die in de vorm geschreven kan worden, met en beide continue functies op het open interval . De algemene oplossing van de bijbehorende homogene differentiaalvergelijking is en een particuliere oplossing is met een willekeurig punt van het domein. Indien constant is (zoals bij een lineair tijdinvariant continu systeem, LTC-systeem, met de tijd) reduceert dit tot het volgende (zie ook eerste-ordesysteem). De algemene oplossing van de bijbehorende homogene differentiaalvergelijking is (exponenti\u00EBle afname of exponenti\u00EBle groei) en de particuliere oplossing met is met een willekeurig punt van het domein. Bij een LTC-systeem is dit op een constante na het outputsignaal bij als inputsignaal, de convolutie van q en de impulsrespons ( vanaf )."@nl . . . "\u041B\u0456\u043D\u0456\u0439\u043D\u0435 \u0434\u0438\u0444\u0435\u0440\u0435\u043D\u0446\u0456\u0430\u043B\u044C\u043D\u0435 \u0440\u0456\u0432\u043D\u044F\u043D\u043D\u044F"@uk . . "Med linj\u00E4r differentialekvation menas en differentialekvation d\u00E4r den s\u00F6kta funktionen och dess derivator endast upptr\u00E4der linj\u00E4rt."@sv . . . . "\u041B\u0438\u043D\u0435\u0439\u043D\u043E\u0435 \u0434\u0438\u0444\u0444\u0435\u0440\u0435\u043D\u0446\u0438\u0430\u043B\u044C\u043D\u043E\u0435 \u0443\u0440\u0430\u0432\u043D\u0435\u043D\u0438\u0435"@ru . . . . . . . "In mathematics, a linear differential equation is a differential equation that is defined by a linear polynomial in the unknown function and its derivatives, that is an equation of the form where a0(x), ..., an(x) and b(x) are arbitrary differentiable functions that do not need to be linear, and y\u2032, ..., y(n) are the successive derivatives of an unknown function y of the variable x. Such an equation is an ordinary differential equation (ODE). A linear differential equation may also be a linear partial differential equation (PDE), if the unknown function depends on several variables, and the derivatives that appear in the equation are partial derivatives. A linear differential equation or a system of linear equations such that the associated homogeneous equations have constant coefficients may be solved by quadrature, which means that the solutions may be expressed in terms of integrals. This is also true for a linear equation of order one, with non-constant coefficients. An equation of order two or higher with non-constant coefficients cannot, in general, be solved by quadrature. For order two, Kovacic's algorithm allows deciding whether there are solutions in terms of integrals, and computing them if any. The solutions of homogeneous linear differential equations with polynomial coefficients are called holonomic functions. This class of functions is stable under sums, products, differentiation, integration, and contains many usual functions and special functions such as exponential function, logarithm, sine, cosine, inverse trigonometric functions, error function, Bessel functions and hypergeometric functions. Their representation by the defining differential equation and initial conditions allows making algorithmic (on these functions) most operations of calculus, such as computation of antiderivatives, limits, asymptotic expansion, and numerical evaluation to any precision, with a certified error bound."@en . . "Equa\u00E7\u00F5es diferenciais lineares s\u00E3o equa\u00E7\u00F5es diferenciais da seguinte forma : As solu\u00E7\u00F5es de uma equa\u00E7\u00E3o diferencial linear podem ser somadas a fim de produzir uma nova solu\u00E7\u00E3o. Diz-se que uma equa\u00E7\u00E3o diferencial \u00E9 linear quando satisfaz duas caracter\u00EDsticas: \n* Cada coeficiente e o termo de n\u00E3o-homogeneidade s\u00F3 dependem da vari\u00E1vel independente, no caso x; \n* A vari\u00E1vel dependente, no caso y, e suas derivadas s\u00E3o de primeiro grau. Um exemplo de equa\u00E7\u00E3o diferencial n\u00E3o linear :"@pt . . . "\u041B\u0456\u043D\u0456\u0439\u043D\u0435 \u0434\u0438\u0444\u0435\u0440\u0435\u043D\u0446\u0456\u0430\u043B\u044C\u043D\u0435 \u0440\u0456\u0432\u043D\u044F\u043D\u043D\u044F \u2014 \u0437\u0432\u0438\u0447\u0430\u0439\u043D\u0435 \u0434\u0438\u0444\u0435\u0440\u0435\u043D\u0446\u0456\u0430\u043B\u044C\u043D\u0435 \u0440\u0456\u0432\u043D\u044F\u043D\u043D\u044F, \u0432 \u044F\u043A\u0435 \u043D\u0435\u0432\u0456\u0434\u043E\u043C\u0430 \u0444\u0443\u043D\u043A\u0446\u0456\u044F \u0442\u0430 \u0457\u0457 \u043F\u043E\u0445\u0456\u0434\u043D\u0456 \u0432\u0445\u043E\u0434\u044F\u0442\u044C \u043B\u0456\u043D\u0456\u0439\u043D\u043E, \u0442\u043E\u0431\u0442\u043E \u0440\u0456\u0432\u043D\u044F\u043D\u043D\u044F \u0432\u0438\u0433\u043B\u044F\u0434\u0443 \u0434\u0435 \u0442\u0430 \u2014 \u0444\u0443\u043D\u043A\u0446\u0456\u0457, \u0449\u043E \u0437\u0430\u043B\u0435\u0436\u0430\u0442\u044C \u0442\u0456\u043B\u044C\u043A\u0438 \u0432\u0456\u0434 \u0430\u0440\u0433\u0443\u043C\u0435\u043D\u0442\u0443 x. \u0412\u0430\u0436\u043B\u0438\u0432\u0438\u0439 \u043F\u0456\u0434\u043A\u043B\u0430\u0441 \u043B\u0456\u043D\u0456\u0439\u043D\u0438\u0445 \u0434\u0438\u0444\u0435\u0440\u0435\u043D\u0446\u0456\u0439\u043D\u0438\u0445 \u0440\u0456\u0432\u043D\u044F\u043D\u044C \u0441\u043A\u043B\u0430\u0434\u0430\u044E\u0442\u044C \u043B\u0456\u043D\u0456\u0439\u043D\u0456 \u0434\u0438\u0444\u0435\u0440\u0435\u043D\u0446\u0456\u0439\u043D\u0456 \u0440\u0456\u0432\u043D\u044F\u043D\u043D\u044F \u0437\u0456 \u0441\u0442\u0430\u043B\u0438\u043C\u0438 \u043A\u043E\u0435\u0444\u0456\u0446\u0456\u0454\u043D\u0442\u0430\u043C\u0438, \u0434\u043B\u044F \u044F\u043A\u0438\u0445 . \u0420\u0456\u0432\u043D\u044F\u043D\u043D\u044F \u043D\u0430\u0437\u0438\u0432\u0430\u0454\u0442\u044C\u0441\u044F \u043E\u0434\u043D\u043E\u0440\u0456\u0434\u043D\u0438\u043C \u043B\u0456\u043D\u0456\u0439\u043D\u0438\u043C \u0434\u0438\u0444\u0435\u0440\u0435\u043D\u0446\u0456\u0439\u043D\u0438\u043C \u0440\u0456\u0432\u043D\u044F\u043D\u043D\u044F\u043C. \u041E\u0434\u043D\u043E\u0440\u0456\u0434\u043D\u0435 \u0434\u0438\u0444\u0435\u0440\u0435\u043D\u0446\u0456\u0439\u043D\u0435 \u0440\u0456\u0432\u043D\u044F\u043D\u043D\u044F n-\u0433\u043E \u043F\u043E\u0440\u044F\u0434\u043A\u0443 \u043C\u0430\u0454 n \u043B\u0456\u043D\u0456\u0439\u043D\u043E \u043D\u0435\u0437\u0430\u043B\u0435\u0436\u043D\u0438\u0445 \u0440\u043E\u0437\u0432'\u044F\u0437\u043A\u0456\u0432. \u042F\u043A\u0449\u043E \u0432\u0456\u0434\u043E\u043C\u0438\u0439 \u0445\u043E\u0447\u0430 \u0431 \u043E\u0434\u0438\u043D \u0447\u0430\u0441\u0442\u043A\u043E\u0432\u0438\u0439 \u0440\u043E\u0437\u0432'\u044F\u0437\u043E\u043A \u043B\u0456\u043D\u0456\u0439\u043D\u043E\u0433\u043E \u0434\u0438\u0444\u0435\u0440\u0435\u043D\u0446\u0456\u0439\u043D\u043E\u0433\u043E \u0440\u0456\u0432\u043D\u044F\u043D\u043D\u044F, \u0442\u043E \u0439\u043E\u0433\u043E \u0437\u0430\u0433\u0430\u043B\u044C\u043D\u0438\u0439 \u0440\u043E\u0437\u0432'\u044F\u0437\u043E\u043A \u0454 \u0441\u0443\u043C\u043E\u044E \u0447\u0430\u0441\u0442\u043A\u043E\u0432\u043E\u0433\u043E \u0440\u043E\u0437\u0432'\u044F\u0437\u043A\u0443 \u0442\u0430 \u043B\u0456\u043D\u0456\u0439\u043D\u043E\u0457 \u043A\u043E\u043C\u0431\u0456\u043D\u0430\u0446\u0456\u0457 n \u0440\u043E\u0437\u0432'\u044F\u0437\u043A\u0456\u0432 \u043E\u0434\u043D\u043E\u0440\u0456\u0434\u043D\u043E\u0433\u043E \u0434\u0438\u0444\u0435\u0440\u0435\u043D\u0446\u0456\u0439\u043D\u043E\u0433\u043E \u0440\u0456\u0432\u043D\u044F\u043D\u043D\u044F."@uk . "\u7EBF\u6027\u5FAE\u5206\u65B9\u7A0B\uFF08\u82F1\u8A9E\uFF1ALinear differential equation\uFF09\u662F\u6570\u5B66\u4E2D\u5E38\u89C1\u7684\u4E00\u7C7B\u5FAE\u5206\u65B9\u7A0B\u3002\u6307\u4EE5\u4E0B\u5F62\u5F0F\u7684\u5FAE\u5206\u65B9\u7A0B\uFF1A \u5176\u4E2D\u65B9\u7A0B\u5DE6\u4FA7\u7684\u5FAE\u5206\u7B97\u5B50\u662F\u7EBF\u6027\u7B97\u5B50\uFF0Cy\u662F\u8981\u89E3\u7684\u672A\u77E5\u51FD\u6570\uFF0C\u65B9\u7A0B\u7684\u53F3\u4FA7\u662F\u4E00\u4E2A\u5DF2\u77E5\u51FD\u6570\u3002\u5982\u679Cf(x) = 0\uFF0C\u90A3\u4E48\u65B9\u7A0B(*)\u7684\u89E3\u7684\u7EBF\u6027\u7EC4\u5408\u4ECD\u7136\u662F\u89E3\uFF0C\u6240\u6709\u7684\u89E3\u6784\u6210\u4E00\u4E2A\u5411\u91CF\u7A7A\u95F4\uFF0C\u79F0\u4E3A\u89E3\u7A7A\u95F4\u3002\u8FD9\u6837\u7684\u65B9\u7A0B\u79F0\u4E3A\u9F50\u6B21\u7EBF\u6027\u5FAE\u5206\u65B9\u7A0B\u3002\u5F53f\u4E0D\u662F\u96F6\u51FD\u6570\u65F6\uFF0C\u6240\u6709\u7684\u89E3\u6784\u6210\u4E00\u4E2A\u4EFF\u5C04\u7A7A\u95F4\uFF0C\u7531\u5BF9\u5E94\u7684\u9F50\u6B21\u65B9\u7A0B\u7684\u89E3\u7A7A\u95F4\u52A0\u4E0A\u4E00\u4E2A\u7279\u89E3\u5F97\u5230\u3002\u8FD9\u6837\u7684\u65B9\u7A0B\u79F0\u4E3A\u975E\u9F50\u6B21\u7EBF\u6027\u5FAE\u5206\u65B9\u7A0B\u3002\u7EBF\u6027\u5FAE\u5206\u65B9\u7A0B\u53EF\u4EE5\u662F\u5E38\u5FAE\u5206\u65B9\u7A0B\uFF0C\u4E5F\u53EF\u4EE5\u662F\u504F\u5FAE\u5206\u65B9\u7A0B\u3002"@zh . . . . . . "In mathematics, a linear differential equation is a differential equation that is defined by a linear polynomial in the unknown function and its derivatives, that is an equation of the form where a0(x), ..., an(x) and b(x) are arbitrary differentiable functions that do not need to be linear, and y\u2032, ..., y(n) are the successive derivatives of an unknown function y of the variable x."@en . . . . . "Equa\u00E7\u00E3o diferencial linear"@pt . . "In matematica, un'equazione differenziale lineare \u00E8 un'equazione differenziale, ordinaria o alle derivate parziali, tale che combinazioni lineari delle sue soluzioni possono essere usate per ottenere altre soluzioni."@it . . . "Linj\u00E4r differentialekvation"@sv . . . . . . "Une \u00E9quation diff\u00E9rentielle lin\u00E9aire est un cas particulier d'\u00E9quation diff\u00E9rentielle pour lequel on peut appliquer des proc\u00E9d\u00E9s de superposition de solutions, et exploiter des r\u00E9sultats d'alg\u00E8bre lin\u00E9aire. De nombreuses \u00E9quations diff\u00E9rentielles de la physique v\u00E9rifient la propri\u00E9t\u00E9 de lin\u00E9arit\u00E9. De plus, les \u00E9quations diff\u00E9rentielles lin\u00E9aires apparaissent naturellement en perturbant une \u00E9quation diff\u00E9rentielle (non lin\u00E9aire) autour d'une de ses solutions. Une \u00E9quation diff\u00E9rentielle lin\u00E9aire scalaire se pr\u00E9sente comme une relation entre une ou plusieurs fonctions inconnues et leurs d\u00E9riv\u00E9es, de la forme o\u00F9 a0, a1, \u2026 an, b sont des fonctions num\u00E9riques continues. Une \u00E9quation diff\u00E9rentielle lin\u00E9aire vectorielle aura le m\u00EAme aspect, en rempla\u00E7ant les ai par des applications lin\u00E9aires (ou souvent des matrices) fonctions de x et b par une fonction de x \u00E0 valeurs vectorielles. Une telle \u00E9quation sera parfois aussi appel\u00E9e syst\u00E8me diff\u00E9rentiel lin\u00E9aire. L'ordre de l'\u00E9quation diff\u00E9rentielle correspond au degr\u00E9 maximal de diff\u00E9rentiation auquel une des fonctions inconnues y a \u00E9t\u00E9 soumise, n dans l'exemple pr\u00E9c\u00E9dent. Il existe des m\u00E9thodes g\u00E9n\u00E9rales de r\u00E9solution pour les \u00E9quations diff\u00E9rentielles lin\u00E9aires scalaires ou ."@fr . . . . . . . "\u00C9quation diff\u00E9rentielle lin\u00E9aire"@fr . . . . . "Equa\u00E7\u00F5es diferenciais lineares s\u00E3o equa\u00E7\u00F5es diferenciais da seguinte forma : As solu\u00E7\u00F5es de uma equa\u00E7\u00E3o diferencial linear podem ser somadas a fim de produzir uma nova solu\u00E7\u00E3o. Diz-se que uma equa\u00E7\u00E3o diferencial \u00E9 linear quando satisfaz duas caracter\u00EDsticas: \n* Cada coeficiente e o termo de n\u00E3o-homogeneidade s\u00F3 dependem da vari\u00E1vel independente, no caso x; \n* A vari\u00E1vel dependente, no caso y, e suas derivadas s\u00E3o de primeiro grau. Um exemplo de equa\u00E7\u00E3o diferencial n\u00E3o linear :"@pt . . "Med linj\u00E4r differentialekvation menas en differentialekvation d\u00E4r den s\u00F6kta funktionen och dess derivator endast upptr\u00E4der linj\u00E4rt."@sv . . "Line\u00E1rn\u00ED diferenci\u00E1ln\u00ED rovnice je diferenci\u00E1ln\u00ED rovnice tvaru kde \n* y je nezn\u00E1m\u00E1 (hledan\u00E1) funkce prom\u011Bnn\u00E9 x, \n* y(k) je k-t\u00E1 derivace funkce y(x), \n* n p\u0159edstavuje \u0159\u00E1d diferenci\u00E1ln\u00ED rovnice, \n* x je nez\u00E1visl\u00E1 prom\u011Bnn\u00E1, \n* ak(x) jsou koeficienty, kter\u00E9 obecn\u011B mohou b\u00FDt funkcemi prom\u011Bnn\u00E9 x. Jsou-li koeficienty ak konstanty, jedn\u00E1 se o diferenci\u00E1ln\u00ED rovnici s konstantn\u00EDmi koeficienty. \n* f(x) p\u0159edstavuje pravou stranu diferenci\u00E1ln\u00ED rovnice. Pokud f(x) = 0, potom se jedn\u00E1 o homogenn\u00ED diferenci\u00E1ln\u00ED rovnici (bez prav\u00E9 strany)."@cs . . . . . . "\u0641\u064A \u0627\u0644\u0631\u064A\u0627\u0636\u064A\u0627\u062A\u060C \u0627\u0644\u0645\u0639\u0627\u062F\u0644\u0629 \u0627\u0644\u062A\u0641\u0627\u0636\u0644\u064A\u0629 \u0627\u0644\u062E\u0637\u064A\u0629 \u0645\u0646 \u0627\u0644\u0631\u062A\u0628\u0629 n \u0647\u064A \u0645\u0639\u0627\u062F\u0644\u0629 \u0645\u0646 \u0627\u0644\u0634\u0643\u0644 \u0627\u0644\u0639\u0627\u0645 \u062D\u064A\u062B \u0648 \u0647\u064A \u062A\u0648\u0627\u0628\u0639 (\u0623\u0648 \u062F\u0627\u0644\u0627\u062A) \u0645\u0639\u0644\u0648\u0645\u0629 \u0648\u062D\u064A\u062B \u060C \u0648 \u0647\u0648 \u062A\u0627\u0628\u0639 \u0645\u062C\u0647\u0648\u0644 \u0648\u0625\u064A\u062C\u0627\u062F \u0647\u0630\u0627 \u0627\u0644\u062A\u0627\u0628\u0639 \u0647\u0648 \u0628\u0645\u062B\u0627\u0628\u0629 \u062D\u0644 \u0644\u0647\u0630\u0647 \u0627\u0644\u0645\u0639\u0627\u062F\u0644\u0629 \u062D\u064A\u062B \u0647\u0646\u0627 \u064A\u0643\u0645\u0646 \u0645\u062D\u0648\u0631 \u0628\u062D\u062B \u0646\u0638\u0631\u064A\u0629 \u0627\u0644\u0645\u0639\u0627\u062F\u0644\u0627\u062A \u0627\u0644\u062A\u0641\u0627\u0636\u0644\u064A\u0629 \u0628\u0634\u0643\u0644 \u0639\u0627\u0645. \u0648\u0639\u0646\u062F\u0645\u0627 \u062A\u0643\u0648\u0646 \u062A\u0633\u0645\u0649 \u0627\u0644\u0645\u0639\u0627\u062F\u0644\u0629 \u062D\u064A\u0646\u0626\u0630\u064D \u0628\u0627\u0644\u0645\u062A\u062C\u0627\u0646\u0633\u0629 Homogeneous \u062D\u064A\u062B \u0625\u064A\u062C\u0627\u062F \u062D\u0644 \u0627\u0644\u0645\u0639\u0627\u062F\u0644\u0629 \u0627\u0644\u0645\u062A\u062C\u0627\u0646\u0633\u0629 \u0647\u0648 \u062E\u0637\u0648\u0629 \u0623\u0648\u0644\u0649 \u0646\u062D\u0648 \u0627\u0644\u062D\u0644 \u0627\u0644\u0639\u0627\u0645 \u0644\u0644\u0645\u0639\u0627\u062F\u0644\u0629 \u0627\u0644\u0644\u0627\u0645\u062A\u062C\u0627\u0646\u0633\u0629 (\u0645\u0641\u0635\u0644 \u0641\u064A \u0627\u0644\u0623\u0633\u0641\u0644). \u0639\u0646\u062F\u0645\u0627 \u062A\u0643\u0648\u0646 \u0627\u0644\u0645\u0639\u0627\u0645\u0644\u0627\u062A \u0645\u062C\u0631\u062F \u0623\u0639\u062F\u0627\u062F \u0646\u0642\u0648\u0644 \u0623\u0646 \u0627\u0644\u0645\u0639\u0627\u062F\u0644\u0629 \u0647\u064A \u0630\u0627\u062A \u0645\u0639\u0627\u0645\u0644\u0627\u062A \u062B\u0627\u0628\u062A\u0629."@ar . . "379868"^^ . . . . . . "\u0641\u064A \u0627\u0644\u0631\u064A\u0627\u0636\u064A\u0627\u062A\u060C \u0627\u0644\u0645\u0639\u0627\u062F\u0644\u0629 \u0627\u0644\u062A\u0641\u0627\u0636\u0644\u064A\u0629 \u0627\u0644\u062E\u0637\u064A\u0629 \u0645\u0646 \u0627\u0644\u0631\u062A\u0628\u0629 n \u0647\u064A \u0645\u0639\u0627\u062F\u0644\u0629 \u0645\u0646 \u0627\u0644\u0634\u0643\u0644 \u0627\u0644\u0639\u0627\u0645 \u062D\u064A\u062B \u0648 \u0647\u064A \u062A\u0648\u0627\u0628\u0639 (\u0623\u0648 \u062F\u0627\u0644\u0627\u062A) \u0645\u0639\u0644\u0648\u0645\u0629 \u0648\u062D\u064A\u062B \u060C \u0648 \u0647\u0648 \u062A\u0627\u0628\u0639 \u0645\u062C\u0647\u0648\u0644 \u0648\u0625\u064A\u062C\u0627\u062F \u0647\u0630\u0627 \u0627\u0644\u062A\u0627\u0628\u0639 \u0647\u0648 \u0628\u0645\u062B\u0627\u0628\u0629 \u062D\u0644 \u0644\u0647\u0630\u0647 \u0627\u0644\u0645\u0639\u0627\u062F\u0644\u0629 \u062D\u064A\u062B \u0647\u0646\u0627 \u064A\u0643\u0645\u0646 \u0645\u062D\u0648\u0631 \u0628\u062D\u062B \u0646\u0638\u0631\u064A\u0629 \u0627\u0644\u0645\u0639\u0627\u062F\u0644\u0627\u062A \u0627\u0644\u062A\u0641\u0627\u0636\u0644\u064A\u0629 \u0628\u0634\u0643\u0644 \u0639\u0627\u0645. \u0648\u0639\u0646\u062F\u0645\u0627 \u062A\u0643\u0648\u0646 \u062A\u0633\u0645\u0649 \u0627\u0644\u0645\u0639\u0627\u062F\u0644\u0629 \u062D\u064A\u0646\u0626\u0630\u064D \u0628\u0627\u0644\u0645\u062A\u062C\u0627\u0646\u0633\u0629 Homogeneous \u062D\u064A\u062B \u0625\u064A\u062C\u0627\u062F \u062D\u0644 \u0627\u0644\u0645\u0639\u0627\u062F\u0644\u0629 \u0627\u0644\u0645\u062A\u062C\u0627\u0646\u0633\u0629 \u0647\u0648 \u062E\u0637\u0648\u0629 \u0623\u0648\u0644\u0649 \u0646\u062D\u0648 \u0627\u0644\u062D\u0644 \u0627\u0644\u0639\u0627\u0645 \u0644\u0644\u0645\u0639\u0627\u062F\u0644\u0629 \u0627\u0644\u0644\u0627\u0645\u062A\u062C\u0627\u0646\u0633\u0629 (\u0645\u0641\u0635\u0644 \u0641\u064A \u0627\u0644\u0623\u0633\u0641\u0644). \u0639\u0646\u062F\u0645\u0627 \u062A\u0643\u0648\u0646 \u0627\u0644\u0645\u0639\u0627\u0645\u0644\u0627\u062A \u0645\u062C\u0631\u062F \u0623\u0639\u062F\u0627\u062F \u0646\u0642\u0648\u0644 \u0623\u0646 \u0627\u0644\u0645\u0639\u0627\u062F\u0644\u0629 \u0647\u064A \u0630\u0627\u062A \u0645\u0639\u0627\u0645\u0644\u0627\u062A \u062B\u0627\u0628\u062A\u0629."@ar . . . . . "Lineare gew\u00F6hnliche Differentialgleichungen bzw. lineare gew\u00F6hnliche Differentialgleichungssysteme sind eine wichtige Klasse von gew\u00F6hnlichen Differentialgleichungen."@de . . . . . "Een lineaire differentiaalvergelijking van eerste orde is een speciaal geval van een lineaire differentiaalvergelijking, die in de vorm geschreven kan worden, met en beide continue functies op het open interval . De algemene oplossing van de bijbehorende homogene differentiaalvergelijking is en een particuliere oplossing is met een willekeurig punt van het domein. Indien constant is (zoals bij een lineair tijdinvariant continu systeem, LTC-systeem, met de tijd) reduceert dit tot het volgende (zie ook eerste-ordesysteem). is (exponenti\u00EBle afname of exponenti\u00EBle groei)"@nl . . "En matem\u00E0tiques, les equacions diferencials lineals s\u00F3n equacions diferencials que tenen solucions que poden sumar-se per obtenir altres solucions. Poden ser ordin\u00E0ries (EDOs) o parcials (EDPs). Les solucions d'equacions linears formen un espai vectorial (a difer\u00E8ncia de les ). Una equaci\u00F3 diferencial lineal \u00E9s una equaci\u00F3 diferencial que t\u00E9 on l'operador diferencial L \u00E9s un operador lineal, y \u00E9s la funci\u00F3 desconeguda (per exemple una funci\u00F3 del temps y(t)), i el terme de la dreta \u0192 \u00E9s una funci\u00F3 donada de la mateixa natura que y. Per a una funci\u00F3 dependent del temps es pot escriure l'equaci\u00F3 com i, fins i tot m\u00E9s precisament L'operador lineal L es pot considerar de la forma. La condici\u00F3 de linealitat de L exclou operacions com el quadrat de la derivada de y; per\u00F2 admet, per exemple, la derivada segona de y.\u00C9s convenient reescriure aquesta equaci\u00F3 en forma d'operador on D \u00E9s l'operador diferencial d/dt (\u00E9s a dir Dy = y, D \u00B2y = y\"... ), i An s\u00F3n funcions donades. Tal equaci\u00F3 es diu que t\u00E9 ordre n, l'\u00EDndex de la derivada m\u00E9s alt de y. Un exemple simple t\u00EDpic \u00E9s l'equaci\u00F3 diferencial lineal que es fa servir per modelitzar la decad\u00E8ncia radioactiva. Sia N(t) el nombre d'\u00E0toms radioactius en alguna mostra de material (com una porci\u00F3 del drap del sudari de Tor\u00ED) en el moment t. Llavors per a alguna constant k > 0,el nombre d'\u00E0toms radioactius que es descomponen es pot modelar per Si y \u00E9s se suposa que \u00E9s una funci\u00F3 de nom\u00E9s una variable, es parla d'una equaci\u00F3 diferencial ordin\u00E0ria, si les derivades i els seus coeficients s'entenen com vectors, matrius o tensors de rang superior, es t\u00E9 una equaci\u00F3 diferencial en derivades parcials (lineal). El cas on \u0192 = 0 s'anomena una equaci\u00F3 homog\u00E8nia i les seves solucions s'anomenen funcions complement\u00E0ries. \u00C9s especialment important per la soluci\u00F3 del cas general, ja que qualsevol funci\u00F3 complement\u00E0ria es pot afegir a una soluci\u00F3 de l'equaci\u00F3 no homog\u00E8nia per donar una altra soluci\u00F3 (per un m\u00E8tode tradicionalment anomenat integral particular i funci\u00F3 complement\u00E0ria). Quan els Ai s\u00F3n nombres, l'equaci\u00F3 es diu que t\u00E9 ."@ca . . . . "En matem\u00E1ticas, una ecuaci\u00F3n diferencial lineal es aquella ecuaci\u00F3n diferencial cuyas soluciones pueden obtenerse mediante combinaciones lineales de otras soluciones. Estas \u00FAltimas pueden ser ordinarias (EDOs) o en derivadas parciales (EDPs). Las soluciones a las ecuaciones diferenciales lineales cuando son homog\u00E9neas forman un espacio vectorial, a diferencia de las ecuaciones diferenciales no lineales."@es . . . . . . . . . . "Une \u00E9quation diff\u00E9rentielle lin\u00E9aire est un cas particulier d'\u00E9quation diff\u00E9rentielle pour lequel on peut appliquer des proc\u00E9d\u00E9s de superposition de solutions, et exploiter des r\u00E9sultats d'alg\u00E8bre lin\u00E9aire. De nombreuses \u00E9quations diff\u00E9rentielles de la physique v\u00E9rifient la propri\u00E9t\u00E9 de lin\u00E9arit\u00E9. De plus, les \u00E9quations diff\u00E9rentielles lin\u00E9aires apparaissent naturellement en perturbant une \u00E9quation diff\u00E9rentielle (non lin\u00E9aire) autour d'une de ses solutions. o\u00F9 a0, a1, \u2026 an, b sont des fonctions num\u00E9riques continues."@fr . . . . . "\u7DDA\u578B\u5FAE\u5206\u65B9\u7A0B\u5F0F\uFF08\u305B\u3093\u3051\u3044\u3073\u3076\u3093\u307B\u3046\u3066\u3044\u3057\u304D\u3001\u82F1: linear differential equation\uFF09\u306F\u3001\u5FAE\u5206\u3092\u7528\u3044\u305F\u7DDA\u578B\u4F5C\u7528\u7D20\uFF08\u7DDA\u578B\u5FAE\u5206\u4F5C\u7528\u7D20\uFF09L \u3068\u672A\u77E5\u95A2\u6570 y \u3068\u65E2\u77E5\u95A2\u6570 b \u3092\u7528\u3044\u3066 Ly = b \u306E\u5F62\u306B\u66F8\u304B\u308C\u308B\u5FAE\u5206\u65B9\u7A0B\u5F0F\u306E\u3053\u3068\u3002"@ja . . "\u041B\u0456\u043D\u0456\u0439\u043D\u0435 \u0434\u0438\u0444\u0435\u0440\u0435\u043D\u0446\u0456\u0430\u043B\u044C\u043D\u0435 \u0440\u0456\u0432\u043D\u044F\u043D\u043D\u044F \u2014 \u0437\u0432\u0438\u0447\u0430\u0439\u043D\u0435 \u0434\u0438\u0444\u0435\u0440\u0435\u043D\u0446\u0456\u0430\u043B\u044C\u043D\u0435 \u0440\u0456\u0432\u043D\u044F\u043D\u043D\u044F, \u0432 \u044F\u043A\u0435 \u043D\u0435\u0432\u0456\u0434\u043E\u043C\u0430 \u0444\u0443\u043D\u043A\u0446\u0456\u044F \u0442\u0430 \u0457\u0457 \u043F\u043E\u0445\u0456\u0434\u043D\u0456 \u0432\u0445\u043E\u0434\u044F\u0442\u044C \u043B\u0456\u043D\u0456\u0439\u043D\u043E, \u0442\u043E\u0431\u0442\u043E \u0440\u0456\u0432\u043D\u044F\u043D\u043D\u044F \u0432\u0438\u0433\u043B\u044F\u0434\u0443 \u0434\u0435 \u0442\u0430 \u2014 \u0444\u0443\u043D\u043A\u0446\u0456\u0457, \u0449\u043E \u0437\u0430\u043B\u0435\u0436\u0430\u0442\u044C \u0442\u0456\u043B\u044C\u043A\u0438 \u0432\u0456\u0434 \u0430\u0440\u0433\u0443\u043C\u0435\u043D\u0442\u0443 x. \u0412\u0430\u0436\u043B\u0438\u0432\u0438\u0439 \u043F\u0456\u0434\u043A\u043B\u0430\u0441 \u043B\u0456\u043D\u0456\u0439\u043D\u0438\u0445 \u0434\u0438\u0444\u0435\u0440\u0435\u043D\u0446\u0456\u0439\u043D\u0438\u0445 \u0440\u0456\u0432\u043D\u044F\u043D\u044C \u0441\u043A\u043B\u0430\u0434\u0430\u044E\u0442\u044C \u043B\u0456\u043D\u0456\u0439\u043D\u0456 \u0434\u0438\u0444\u0435\u0440\u0435\u043D\u0446\u0456\u0439\u043D\u0456 \u0440\u0456\u0432\u043D\u044F\u043D\u043D\u044F \u0437\u0456 \u0441\u0442\u0430\u043B\u0438\u043C\u0438 \u043A\u043E\u0435\u0444\u0456\u0446\u0456\u0454\u043D\u0442\u0430\u043C\u0438, \u0434\u043B\u044F \u044F\u043A\u0438\u0445 . \u0420\u0456\u0432\u043D\u044F\u043D\u043D\u044F \u043D\u0430\u0437\u0438\u0432\u0430\u0454\u0442\u044C\u0441\u044F \u043E\u0434\u043D\u043E\u0440\u0456\u0434\u043D\u0438\u043C \u043B\u0456\u043D\u0456\u0439\u043D\u0438\u043C \u0434\u0438\u0444\u0435\u0440\u0435\u043D\u0446\u0456\u0439\u043D\u0438\u043C \u0440\u0456\u0432\u043D\u044F\u043D\u043D\u044F\u043C. \u041E\u0434\u043D\u043E\u0440\u0456\u0434\u043D\u0435 \u0434\u0438\u0444\u0435\u0440\u0435\u043D\u0446\u0456\u0439\u043D\u0435 \u0440\u0456\u0432\u043D\u044F\u043D\u043D\u044F n-\u0433\u043E \u043F\u043E\u0440\u044F\u0434\u043A\u0443 \u043C\u0430\u0454 n \u043B\u0456\u043D\u0456\u0439\u043D\u043E \u043D\u0435\u0437\u0430\u043B\u0435\u0436\u043D\u0438\u0445 \u0440\u043E\u0437\u0432'\u044F\u0437\u043A\u0456\u0432."@uk . "In matematica, un'equazione differenziale lineare \u00E8 un'equazione differenziale, ordinaria o alle derivate parziali, tale che combinazioni lineari delle sue soluzioni possono essere usate per ottenere altre soluzioni."@it . . . . . . . . . "29998"^^ . . . . "En matem\u00E1ticas, una ecuaci\u00F3n diferencial lineal es aquella ecuaci\u00F3n diferencial cuyas soluciones pueden obtenerse mediante combinaciones lineales de otras soluciones. Estas \u00FAltimas pueden ser ordinarias (EDOs) o en derivadas parciales (EDPs). Las soluciones a las ecuaciones diferenciales lineales cuando son homog\u00E9neas forman un espacio vectorial, a diferencia de las ecuaciones diferenciales no lineales."@es . . . . . . "Lineare gew\u00F6hnliche Differentialgleichungen bzw. lineare gew\u00F6hnliche Differentialgleichungssysteme sind eine wichtige Klasse von gew\u00F6hnlichen Differentialgleichungen."@de . . . . . . . . . . . "Line\u00E1rn\u00ED diferenci\u00E1ln\u00ED rovnice je diferenci\u00E1ln\u00ED rovnice tvaru kde \n* y je nezn\u00E1m\u00E1 (hledan\u00E1) funkce prom\u011Bnn\u00E9 x, \n* y(k) je k-t\u00E1 derivace funkce y(x), \n* n p\u0159edstavuje \u0159\u00E1d diferenci\u00E1ln\u00ED rovnice, \n* x je nez\u00E1visl\u00E1 prom\u011Bnn\u00E1, \n* ak(x) jsou koeficienty, kter\u00E9 obecn\u011B mohou b\u00FDt funkcemi prom\u011Bnn\u00E9 x. Jsou-li koeficienty ak konstanty, jedn\u00E1 se o diferenci\u00E1ln\u00ED rovnici s konstantn\u00EDmi koeficienty. \n* f(x) p\u0159edstavuje pravou stranu diferenci\u00E1ln\u00ED rovnice. Pokud f(x) = 0, potom se jedn\u00E1 o homogenn\u00ED diferenci\u00E1ln\u00ED rovnici (bez prav\u00E9 strany). V line\u00E1rn\u00ED diferenci\u00E1ln\u00ED rovnici se hledan\u00E1 funkce vyskytuje pouze line\u00E1rn\u011B a nikde se nevyskytuj\u00ED sou\u010Diny hledan\u00E9 funkce s jej\u00EDmi derivacemi, ani sou\u010Diny derivac\u00ED t\u00E9to funkce. Line\u00E1rn\u00ED diferenci\u00E1ln\u00ED rovnice mohou b\u00FDt oby\u010Dejn\u00E9 (s jednou nez\u00E1vislou prom\u011Bnnou) i parci\u00E1ln\u00ED (s v\u00EDce nez\u00E1visl\u00FDmi prom\u011Bnn\u00FDmi). \u0158e\u0161en\u00ED line\u00E1rn\u00ED rovnice tvo\u0159\u00ED (na rozd\u00EDl od \u0159e\u0161en\u00ED neline\u00E1rn\u00EDch diferenci\u00E1ln\u00EDch rovnic) vektorov\u00FD prostor."@cs . . . . "Equazione differenziale lineare"@it . . . . . . . "\u7DDA\u578B\u5FAE\u5206\u65B9\u7A0B\u5F0F\uFF08\u305B\u3093\u3051\u3044\u3073\u3076\u3093\u307B\u3046\u3066\u3044\u3057\u304D\u3001\u82F1: linear differential equation\uFF09\u306F\u3001\u5FAE\u5206\u3092\u7528\u3044\u305F\u7DDA\u578B\u4F5C\u7528\u7D20\uFF08\u7DDA\u578B\u5FAE\u5206\u4F5C\u7528\u7D20\uFF09L \u3068\u672A\u77E5\u95A2\u6570 y \u3068\u65E2\u77E5\u95A2\u6570 b \u3092\u7528\u3044\u3066 Ly = b \u306E\u5F62\u306B\u66F8\u304B\u308C\u308B\u5FAE\u5206\u65B9\u7A0B\u5F0F\u306E\u3053\u3068\u3002"@ja .