# Численное решение дифференциальных уравнений В задачах

```&times;&egrave;&ntilde;&euml;&aring;&iacute;&iacute;&icirc;&aring; &eth;&aring;&oslash;&aring;&iacute;&egrave;&aring; &auml;&egrave;&ocirc;&ocirc;&aring;&eth;&aring;&iacute;&ouml;&egrave;&agrave;&euml;&uuml;&iacute;&ucirc;&otilde; &oacute;&eth;&agrave;&acirc;&iacute;&aring;&iacute;&egrave;&eacute;
&Acirc; &ccedil;&agrave;&auml;&agrave;&divide;&agrave;&otilde; &igrave;&aring;&otilde;&agrave;&iacute;&egrave;&ecirc;&egrave; &divide;&agrave;&ntilde;&ograve;&icirc; &acirc;&icirc;&ccedil;&iacute;&egrave;&ecirc;&agrave;&thorn;&ograve; &auml;&egrave;&ocirc;&ocirc;&aring;&eth;&aring;&iacute;&ouml;&egrave;&agrave;&euml;&uuml;&iacute;&ucirc;&aring; &oacute;&eth;&agrave;&acirc;&iacute;&aring;&iacute;&egrave;&yuml;,
&ecirc;&icirc;&ograve;&icirc;&eth;&ucirc;&aring; &iacute;&aring;&euml;&uuml;&ccedil;&yuml; &eth;&aring;&oslash;&egrave;&ograve;&uuml; &agrave;&iacute;&agrave;&euml;&egrave;&ograve;&egrave;&divide;&aring;&ntilde;&ecirc;&egrave;, &ograve;. &aring;. &iuml;&icirc;&euml;&oacute;&divide;&egrave;&ograve;&uuml; &yuml;&acirc;&iacute;&oacute;&thorn; &ocirc;&icirc;&eth;&igrave;&oacute;&euml;&oacute; &auml;&euml;&yuml;
&eth;&aring;&oslash;&aring;&iacute;&egrave;&yuml;, &iuml;&oacute;&ntilde;&ograve;&uuml; &auml;&agrave;&aelig;&aring; &ntilde; &egrave;&ntilde;&iuml;&icirc;&euml;&uuml;&ccedil;&icirc;&acirc;&agrave;&iacute;&egrave;&aring;&igrave; &ntilde;&egrave;&igrave;&acirc;&icirc;&euml;&agrave; &egrave;&iacute;&ograve;&aring;&atilde;&eth;&agrave;&euml;&agrave;. &Ograve;&agrave;&ecirc;&egrave;&aring; &auml;&egrave;&ocirc;&ocirc;&aring;&eth;&aring;&iacute;&ouml;&egrave;&agrave;&euml;&uuml;&iacute;&ucirc;&aring; &oacute;&eth;&agrave;&acirc;&iacute;&aring;&iacute;&egrave;&yuml; &igrave;&icirc;&aelig;&iacute;&icirc; &eth;&aring;&oslash;&egrave;&ograve;&uuml; &ograve;&icirc;&euml;&uuml;&ecirc;&icirc; &divide;&egrave;&ntilde;&euml;&aring;&iacute;&iacute;&icirc;, &ograve;. &aring;. &iuml;&eth;&egrave;&aacute;&euml;&egrave;&aelig;&aring;&iacute;&iacute;&icirc;
&iuml;&eth;&egrave; &iuml;&icirc;&igrave;&icirc;&ugrave;&egrave; &icirc;&iuml;&eth;&aring;&auml;&aring;&euml;&aring;&iacute;&iacute;&ucirc;&otilde; &acirc;&ucirc;&divide;&egrave;&ntilde;&euml;&aring;&iacute;&egrave;&eacute;.
&Ccedil;&auml;&aring;&ntilde;&uuml; &aacute;&oacute;&auml;&aring;&ograve; &icirc;&iuml;&egrave;&ntilde;&agrave;&iacute; &icirc;&auml;&egrave;&iacute; &egrave;&ccedil; &ntilde;&agrave;&igrave;&ucirc;&otilde; &iuml;&icirc;&iuml;&oacute;&euml;&yuml;&eth;&iacute;&ucirc;&otilde; &igrave;&aring;&ograve;&icirc;&auml;&icirc;&acirc; &auml;&euml;&yuml; &divide;&egrave;&ntilde;&euml;&aring;&iacute;&iacute;&icirc;&atilde;&icirc;
&eth;&aring;&oslash;&aring;&iacute;&egrave;&yuml; &auml;&egrave;&ocirc;&ocirc;&aring;&eth;&aring;&iacute;&ouml;&egrave;&agrave;&euml;&uuml;&iacute;&ucirc;&otilde; &oacute;&eth;&agrave;&acirc;&iacute;&aring;&iacute;&egrave;&eacute; &egrave; &ntilde;&egrave;&ntilde;&ograve;&aring;&igrave; &iuml;&aring;&eth;&acirc;&icirc;&atilde;&icirc; &iuml;&icirc;&eth;&yuml;&auml;&ecirc;&agrave;. &Acirc;&icirc;&icirc;&aacute;&ugrave;&aring;
&atilde;&icirc;&acirc;&icirc;&eth;&yuml;, &acirc; &igrave;&aring;&otilde;&agrave;&iacute;&egrave;&ecirc;&aring; &icirc;&aacute;&ucirc;&divide;&iacute;&icirc; &acirc;&ntilde;&ograve;&eth;&aring;&divide;&agrave;&thorn;&ograve;&ntilde;&yuml; &oacute;&eth;&agrave;&acirc;&iacute;&aring;&iacute;&egrave;&yuml; &egrave; &ntilde;&egrave;&ntilde;&ograve;&aring;&igrave;&ucirc; &oacute;&eth;&agrave;&acirc;&iacute;&aring;&iacute;&egrave;&eacute;
&acirc;&ograve;&icirc;&eth;&icirc;&atilde;&icirc; &iuml;&icirc;&eth;&yuml;&auml;&ecirc;&agrave; (&oacute;&ntilde;&ecirc;&icirc;&eth;&aring;&iacute;&egrave;&aring; &acirc;&ucirc;&eth;&agrave;&aelig;&agrave;&aring;&ograve;&ntilde;&yuml; &divide;&aring;&eth;&aring;&ccedil; &iuml;&icirc;&euml;&icirc;&aelig;&aring;&iacute;&egrave;&aring; &egrave; &ntilde;&ecirc;&icirc;&eth;&icirc;&ntilde;&ograve;&uuml;), &iuml;&icirc;&yacute;&ograve;&icirc;&igrave;&oacute; &iuml;&aring;&eth;&aring;&auml; &iuml;&eth;&egrave;&igrave;&aring;&iacute;&aring;&iacute;&egrave;&aring;&igrave; &yacute;&ograve;&icirc;&atilde;&icirc; &igrave;&aring;&ograve;&icirc;&auml;&agrave; &egrave;&otilde; &iacute;&oacute;&aelig;&iacute;&icirc; &iuml;&eth;&aring;&icirc;&aacute;&eth;&agrave;&ccedil;&icirc;&acirc;&agrave;&ograve;&uuml; &ecirc; &ntilde;&egrave;&ntilde;&ograve;&aring;&igrave;&agrave;&igrave;
&oacute;&eth;&agrave;&acirc;&iacute;&aring;&iacute;&egrave;&eacute; &iuml;&aring;&eth;&acirc;&icirc;&atilde;&icirc; &iuml;&icirc;&eth;&yuml;&auml;&ecirc;&agrave;, &acirc;&acirc;&aring;&auml;&yuml; &auml;&icirc;&iuml;&icirc;&euml;&iacute;&egrave;&ograve;&aring;&euml;&uuml;&iacute;&ucirc;&aring; &iuml;&aring;&eth;&aring;&igrave;&aring;&iacute;&iacute;&ucirc;&aring;, &icirc;&aacute;&icirc;&ccedil;&iacute;&agrave;&divide;&agrave;&thorn;&ugrave;&egrave;&aring; &iuml;&aring;&eth;&acirc;&ucirc;&aring; &iuml;&eth;&icirc;&egrave;&ccedil;&acirc;&icirc;&auml;&iacute;&ucirc;&aring; &ecirc;&icirc;&icirc;&eth;&auml;&egrave;&iacute;&agrave;&ograve;.
&Auml;&agrave;&iacute;&iacute;&ucirc;&eacute; &igrave;&aring;&ograve;&icirc;&auml; (&igrave;&aring;&ograve;&icirc;&auml; &ETH;&oacute;&iacute;&atilde;&aring;-&Ecirc;&oacute;&ograve;&ograve;&ucirc; 4 &iuml;&icirc;&eth;&yuml;&auml;&ecirc;&agrave;), &ecirc;&agrave;&ecirc; &egrave; &aacute;&icirc;&euml;&uuml;&oslash;&icirc;&aring; &ecirc;&icirc;&euml;&egrave;&divide;&aring;&ntilde;&ograve;&acirc;&icirc; &auml;&eth;&oacute;&atilde;&egrave;&otilde; &divide;&egrave;&ntilde;&euml;&aring;&iacute;&iacute;&ucirc;&otilde; &igrave;&aring;&ograve;&icirc;&auml;&icirc;&acirc; &auml;&euml;&yuml; &eth;&aring;&oslash;&aring;&iacute;&egrave;&eacute; &ccedil;&agrave;&auml;&agrave;&divide;&egrave; &Ecirc;&icirc;&oslash;&egrave; &auml;&euml;&yuml; &oacute;&eth;&agrave;&acirc;&iacute;&aring;&iacute;&egrave;&eacute;
&egrave; &ntilde;&egrave;&ntilde;&ograve;&aring;&igrave; &iuml;&aring;&eth;&acirc;&icirc;&atilde;&icirc; &iuml;&icirc;&eth;&yuml;&auml;&ecirc;&agrave;, &egrave;&igrave;&aring;&aring;&ograve; &auml;&aring;&euml;&icirc; &ntilde; &ccedil;&agrave;&auml;&agrave;&divide;&agrave;&igrave;&egrave; &Ecirc;&icirc;&oslash;&egrave; (&ccedil;&agrave;&auml;&agrave;&iacute;&icirc; &iacute;&agrave;&divide;&agrave;&euml;&uuml;&iacute;&icirc;&aring;
&ccedil;&iacute;&agrave;&divide;&aring;&iacute;&egrave;&aring; &egrave;&ntilde;&ecirc;&icirc;&igrave;&icirc;&eacute; &ocirc;&oacute;&iacute;&ecirc;&ouml;&egrave;&egrave;) &acirc;&egrave;&auml;&agrave;:
dy
= F (x, y), y(x0 ) = y0 .
dx
&Ccedil;&auml;&aring;&ntilde;&uuml; &igrave;&ucirc; &iacute;&aring; &aacute;&oacute;&auml;&aring;&igrave; &auml;&aring;&euml;&agrave;&ograve;&uuml; &iacute;&egrave;&ecirc;&agrave;&ecirc;&icirc;&atilde;&icirc; &eth;&agrave;&ccedil;&euml;&egrave;&divide;&egrave;&yuml; &igrave;&aring;&aelig;&auml;&oacute; &oacute;&eth;&agrave;&acirc;&iacute;&aring;&iacute;&egrave;&yuml;&igrave;&egrave; &egrave; &ntilde;&egrave;&ntilde;&ograve;&aring;&igrave;&agrave;&igrave;&egrave; &oacute;&eth;&agrave;&acirc;&iacute;&aring;&iacute;&egrave;&eacute;, &iuml;&eth;&icirc;&ntilde;&ograve;&icirc; &acirc; &ntilde;&euml;&oacute;&divide;&agrave;&aring; &oacute;&eth;&agrave;&acirc;&iacute;&aring;&iacute;&egrave;&yuml; y(x) &aacute;&oacute;&auml;&aring;&ograve; &icirc;&aacute;&icirc;&ccedil;&iacute;&agrave;&divide;&agrave;&ograve;&uuml; &ntilde;&ecirc;&agrave;&euml;&yuml;&eth;&iacute;&oacute;&thorn; &ocirc;&oacute;&iacute;&ecirc;&ouml;&egrave;&thorn;, &agrave; &acirc; &ntilde;&euml;&oacute;&divide;&agrave;&aring; &ntilde;&egrave;&ntilde;&ograve;&aring;&igrave; &acirc;&aring;&ecirc;&ograve;&icirc;&eth;-&ocirc;&oacute;&iacute;&ecirc;&ouml;&egrave;&thorn; &egrave;&ccedil; &ntilde;&icirc;&icirc;&ograve;&acirc;&aring;&ograve;&ntilde;&ograve;&acirc;&oacute;&thorn;&ugrave;&aring;&atilde;&icirc; &divide;&egrave;&ntilde;&euml;&agrave; &ecirc;&icirc;&igrave;&iuml;&icirc;&iacute;&aring;&iacute;&ograve;.
&Igrave;&aring;&ograve;&icirc;&auml; &ntilde;&icirc;&ntilde;&ograve;&icirc;&egrave;&ograve; &acirc; &iuml;&icirc;&ntilde;&euml;&aring;&auml;&icirc;&acirc;&agrave;&ograve;&aring;&euml;&uuml;&iacute;&icirc;&igrave; &acirc;&ucirc;&divide;&egrave;&ntilde;&euml;&aring;&iacute;&egrave;&egrave; &iuml;&eth;&egrave;&aacute;&euml;&egrave;&aelig;&aring;&iacute;&iacute;&ucirc;&otilde; &ccedil;&iacute;&agrave;&divide;&aring;&iacute;&egrave;&eacute; yi &auml;&euml;&yuml; &egrave;&ntilde;&ecirc;&icirc;&igrave;&ucirc;&otilde; &acirc;&aring;&euml;&egrave;&divide;&egrave;&iacute; y(xi ), &atilde;&auml;&aring; xi &iuml;&icirc;&ntilde;&euml;&aring;&auml;&icirc;&acirc;&agrave;&ograve;&aring;&euml;&uuml;&iacute;&icirc;&ntilde;&ograve;&uuml; &ograve;&icirc;&divide;&aring;&ecirc;, &acirc;
&ecirc;&icirc;&ograve;&icirc;&eth;&ucirc;&otilde; &iacute;&agrave;&igrave; &iacute;&oacute;&aelig;&iacute;&icirc; &iuml;&icirc;&euml;&oacute;&divide;&egrave;&ograve;&uuml; &iuml;&eth;&egrave;&aacute;&euml;&egrave;&aelig;&aring;&iacute;&iacute;&ucirc;&aring; &ccedil;&iacute;&agrave;&divide;&aring;&iacute;&egrave;&yuml; &egrave;&ntilde;&ecirc;&icirc;&igrave;&icirc;&eacute; &ocirc;&oacute;&iacute;&ecirc;&ouml;&egrave;&egrave;.
&Acirc; &iuml;&eth;&icirc;&ntilde;&ograve;&aring;&eacute;&oslash;&aring;&igrave; &ntilde;&euml;&oacute;&divide;&agrave;&aring; (&aacute;&icirc;&euml;&aring;&aring; &ntilde;&euml;&icirc;&aelig;&iacute;&ucirc;&aring; &acirc;&agrave;&eth;&egrave;&agrave;&iacute;&ograve;&ucirc; &igrave;&ucirc; &eth;&agrave;&ntilde;&ntilde;&igrave;&agrave;&ograve;&eth;&egrave;&acirc;&agrave;&ograve;&uuml; &iacute;&aring; &aacute;&oacute;&auml;&aring;&igrave;) xi &agrave;&eth;&egrave;&ocirc;&igrave;&aring;&ograve;&egrave;&divide;&aring;&ntilde;&ecirc;&agrave;&yuml; &iuml;&eth;&icirc;&atilde;&eth;&aring;&ntilde;&ntilde;&egrave;&yuml; &ntilde; &eth;&agrave;&ccedil;&iacute;&icirc;&ntilde;&ograve;&uuml;&thorn; h (&yacute;&ograve;&icirc; &divide;&egrave;&ntilde;&euml;&icirc; &iacute;&agrave;&ccedil;&ucirc;&acirc;&agrave;&aring;&ograve;&ntilde;&yuml;
&oslash;&agrave;&atilde;&icirc;&igrave; &ntilde;&aring;&ograve;&ecirc;&egrave;, &agrave; &ntilde;&agrave;&igrave;&egrave; xi &aring;&aring; &oacute;&ccedil;&euml;&agrave;&igrave;&egrave;).
&ETH;&agrave;&ntilde;&divide;&aring;&ograve;&iacute;&ucirc;&aring; &ocirc;&icirc;&eth;&igrave;&oacute;&euml;&ucirc; &auml;&euml;&yuml; &auml;&agrave;&iacute;&iacute;&icirc;&atilde;&icirc; &igrave;&aring;&ograve;&icirc;&auml;&agrave; &acirc;&ucirc;&atilde;&euml;&yuml;&auml;&yuml;&ograve; &ntilde;&euml;&aring;&auml;&oacute;&thorn;&ugrave;&egrave;&igrave; &icirc;&aacute;&eth;&agrave;&ccedil;&icirc;&igrave;:
&atilde;&auml;&aring;
1
yn+1 = yn + (k1 + 2k2 + 2k3 + k4 ),
6
k1 = hF (xn , yn ),
h
1
k2 = hF (xn + , yn + k1 ),
2
2
h
1
k3 = hF (xn + , yn + k2 ),
2
2
k4 = hF (xn + h, yn + k3 ).
(1)
(2)
(3)
(4)
&Icirc;&ouml;&aring;&iacute;&ecirc;&agrave; &iuml;&icirc;&atilde;&eth;&aring;&oslash;&iacute;&icirc;&ntilde;&ograve;&egrave; &auml;&agrave;&iacute;&iacute;&icirc;&atilde;&icirc; &igrave;&aring;&ograve;&icirc;&auml;&agrave; &auml;&icirc;&acirc;&icirc;&euml;&uuml;&iacute;&icirc; &ograve;&eth;&oacute;&auml;&iacute;&icirc;&aring; &auml;&aring;&euml;&icirc;, &icirc;&auml;&iacute;&agrave;&ecirc;&icirc;
&egrave;&igrave;&aring;&aring;&ograve;&ntilde;&yuml; &ntilde;&euml;&aring;&auml;&oacute;&thorn;&ugrave;&agrave;&yuml; &ocirc;&icirc;&eth;&igrave;&oacute;&euml;&agrave; &auml;&euml;&yuml; &atilde;&euml;&agrave;&acirc;&iacute;&icirc;&atilde;&icirc; &divide;&euml;&aring;&iacute;&agrave; &agrave;&ntilde;&egrave;&igrave;&iuml;&ograve;&icirc;&ograve;&egrave;&ecirc;&egrave; &iuml;&icirc;&atilde;&eth;&aring;&oslash;&iacute;&icirc;&ntilde;&ograve;&egrave;
1
&iacute;&agrave; &icirc;&divide;&aring;&eth;&aring;&auml;&iacute;&icirc;&igrave; &oslash;&agrave;&atilde;&aring;, &acirc; &iuml;&eth;&aring;&auml;&iuml;&icirc;&euml;&icirc;&aelig;&aring;&iacute;&egrave;&egrave;, &divide;&ograve;&icirc; &iuml;&eth;&aring;&auml;&ucirc;&auml;&oacute;&ugrave;&egrave;&aring; &oslash;&agrave;&atilde;&egrave; &acirc;&ucirc;&divide;&egrave;&ntilde;&euml;&aring;&iacute;&ucirc;
&ograve;&icirc;&divide;&iacute;&icirc;:
1
y(xn+1 ) − yn+1 ≈
31
∗
(yn+1 − yn+1
),
∗
&atilde;&auml;&aring; yn+1
&iuml;&eth;&egrave;&aacute;&euml;&egrave;&aelig;&aring;&iacute;&iacute;&icirc;&aring; &ccedil;&iacute;&agrave;&divide;&aring;&iacute;&egrave;&aring; &egrave;&ntilde;&ecirc;&icirc;&igrave;&icirc;&eacute; &ocirc;&oacute;&iacute;&ecirc;&ouml;&egrave;&egrave;, &iuml;&icirc;&euml;&oacute;&divide;&aring;&iacute;&iacute;&icirc;&aring; &iuml;&icirc; yn−1
&egrave; xn−1 &ntilde; &oslash;&agrave;&atilde;&icirc;&igrave; 2h.
2
```