\section{‚cY¤Y?Y.}
„ i a !®a a®¤Ya¦?a aYeY?Y a §«?cea ?a Ycea § ¤ c, ®!§®a ¬Ya®¤®c
?a aYeY?i, a ?¦Y Y¬®F® aY®a??. ‚ YY a ?¦Y c?«icYe aY®aYa?cYa??Y
a §¤Y«e, ?®acieYeY «?Y©e¬ aa cY?i¬ a ?®aa®ie¬? ? ?YaY¬Ye¬?
?®iaeae?ae?Ya ¬?, Y ®a®aie?Yai ?ai¬e¬ ®!a §®¬ ? ?a Yce¬ § ¤ c ¬.
˜a® !e«® Y®!a®¤?¬® ¤«i a®F®, ca®!e ?§«®¦Ye© ¬ aYa? « !e« ¤®aa a®cY
¤«i ?®?¬ ?i ¬Ya®¤®c aYeY?i.
˜ !®a a®aa®?a ?§ ¤caa a §¤Y«®c:
\begin{description}
\item[˜Y¬®F® aY®a??.]
?cYi ?a a?®Y ?§«®¦Y?Y a?®a®!®c aYeY?i ?a Ycea § ¤ c, a ?¦Y ¤aaF?Y
a??e § ¤ c, Y®!a®¤?¬eY ¤«i aa?Ye®F® ®ac®Y?i ¬ aYa? « .
\item[˜YeY?Y § ¤ c.]
˜YeY?Y a §«?cea a??®c § ¤ c ?a YceY aa«®c?i (?a Ycea § ¤ c).
\end{description}
\section {˜Y¬®F® aY®a??.}
\subsection{‹?Y©eY aa cY?i a ?®aa®ie¬? ?®iaeae?ae?Ya ¬?.}
\begin{enumerate}
\item —a®!e aYe?ai «?Y©®Y ®¤®a®¤®Y aa cY?Y a ?®aa®ie¬? ?®iaeae?ae?Ya ¬?
$$
a_0y^{n}+a_1y^{n-1}+…+a_{n-1}y’+a_ny=0, \eqno{(1)}
$$
¤® a®aa c?ai a a ?aYa?aa?cYa?®Y aa cY?Y
$$
a_0\lambda^n+a_1\lambda^{n-1}+…+a_{n-1}\lambda+a_n=0 \eqno{(2)}
$$
? ©a? caY YF® ?®a? $\lambda_1,…,\lambda_n$.
?!eYY aYeY?Y aa cY?i $(1)$ Yaai aa¬¬ , a®aa®ie i ?§ a« F Y¬ea c?¤
$C_ie^{\lambda_i x}$ ¤«i ? ¦¤®F® ?a®aa®F® ?®ai $\lambda_i$ aa cY?i $(2)$
? a« F Y¬ea c?¤
$$
(C_{m+1}+C_{m+2}x+C_{m+3}x^2+…+C_{m+k}x^{k-1})e^{\lambda x}
$$
¤«i ? ¦¤®F® ?a a®F® ?®ai $\lambda$ aa cY?i $(2)$, F¤Y $k$-?a a®aai
?®ai. ‚aY $C_i$- ?a®?§c®«ieY ?®aa®ieY. ?®iaeae?ae?Yae aa cY?i $(1)$ ?
?®a? $\lambda$ §¤Yai ¬®Faa !eai cYeYaacYe¬? ?«? ?®¬?«Y?ae¬?.
…a«? ¦Y caY ?®iaeae?ae?Yae aa cY?i $(1)$ cYeYaacYeY, a® aYeY?Y ¬®¦®
§ ??a ai c cYeYaacY®© ae®a¬Y ? c a«ac Y ?®¬?«Y?aea ?®aY© $\lambda$.
„«i ? ¦¤®© ? ae ?®¬?«Y?aea a®?ai¦Yea ?®aY© $\lambda=\alpha\pm\beta i$
c ae®a¬a«a ®!eYF® aYeY?i c?«ic iaai a« F Y¬eY
$$
C_{m+1}e^{\alpha x}\cos{\beta x}+C_{m+2}e^{\alpha x}\sin{\beta x},
$$
Ya«? ia? ?®a? ?a®aaeY, ? a« F Y¬eY
$$
P_{k-1}(x)e^{\alpha x}\cos{\beta x}+Q_{k-1}e^{\alpha x}\sin{\beta x}
$$
Ya«? ? ¦¤e© ?§ ?®aY© $\alpha+\beta i$ ? $\alpha-\beta i$ ?¬YYa
?a a®aai $k$. ‡¤Yai $P_{k-1}$ ? $Q_{k-1}$- ¬®F®c«Ye aaY?Y? $k-1$,
«®F?ceY ¬®F®c«Ya c $(3)$, ?a ?®iaeae?ae?Yae- ?a®?§c®«ieY ?®aa®ieY.
\item ‹?Y©®Y Y®¤®a®¤®Y aa cY?Y
$$
a_0y^{n}+a_1y^{n-1}+…+a_ny=f(x) \eqno{(11)}
$$
a «i!®© ?a c®© c aaii $f(x)$ aYe Yaai ¬Ya®¤®¬ c a? ae?? ?®aa®iea. ˜aaai
©¤Y® aYeY?Y $y=C_1y_1+…+C_ny_n$ «?Y©®F® ®¤®a®¤®F® aa cY?i a
a®© ¦Y «Yc®© c aaii. ’®F¤ aYeY?Y $(1)$ ?eYaai c c?¤Y
$$
y=C_1(x)y_1+…+C_n(x)y_n
$$
”a?ae?? $C_i(x)$ ®?aY¤Y«iiaai ?§ a?aaY¬e
$$
\left\{
\begin{array}{l}
C_1’y_1+…+C_n’y_n=0,\\
C_1’y_1’+…+C_n’y_n’=0,\\
……………………………..\\
C_1’y_1^{(n-2)}+…+C_n’y_n^{n-2}=0,\\
a_0(C_1’y_1^{(n-1)}+…+C_n’y_n^{n-1})=f(x).
\end{array}
\right.
$$
\end{enumerate}
\subsection{‹?Y©eY aa cY?i a ?YaY¬Ye¬? ?®iaeae?ae?Ya ¬?.}\label{perem}
\begin{enumerate}
\item …a«? ?§cYaa® c aa®Y aYeY?Y $y_1$ «?Y©®F® ®¤®a®¤®F® aa cY?i
n-F® ?®ai¤? , a® ?®ai¤®? ¬®¦® ?®?§?ai, a®aa ii «?Y©®aai aa cY?i.
„«i ia®F® c aa cY?Y ¤® ?®¤aa c?ai $y=y_1z$ ? § aY¬ ?®?§?ai ?®ai¤®? § ¬Y®©
$z’=u$.
—a®!e ©a? ®!eYY aYeY?Y «?Y©®F® ®¤®a®¤®F® aa cY?i ca®a®F® ?®ai¤?
$a_0(x)y”+a_1(x)y’+a_2(x)y=0$, a ?®a®a®F® ?§cYaa® ®¤® c aa®Y aYeY?Y
$y_1$, ¬®¦® ?®?§?ai ?®ai¤®? aa cY?i a? § e¬ ceeY a?®a®!®¬. ?¤ ?®
a¤®!YY c®a?®«i§®c aiai ae®a¬a«®© ?aaa®Fa ¤a?®F®-‹?ac?««i\label{ostrog}:
$$
\left|
\begin{array}{l}
y_1, y_2\\
y_1′, y_2′
\end{array}
\right|=Ce^{-\int{\rho(x)dx}},\rho(x)=\frac{a_1(x)}{a_0(x)},
$$
F¤Y $y_1$ ? $y_2$-«i!eY ¤c aYeY?i ¤ ®F® aa cY?i.
\item ?!eYF® ¬Ya®¤ ¤«i ®aea? ?i c aa®F® aYeY?i «?Y©®F® aa cY?i ca®a®F®
?®ai¤? Y aaeYaacaYa. ‚ Y?®a®aea a«ac ia aYeY?Y a¤ Yaai ©a? a ?®¬®eii
?®¤!®a .
{\it ˜a?¬Ya.} \label{prim}˜ ©a? c aa®Y aYeY?Y aa cY?i
$$
(1-2x^2)y”+2y’+4y=0, \eqno{(2)}
$$
ic«iieYYai «FY!a ?cYa??¬ ¬®F®c«Y®¬ (Ya«? a ?®Y aYeY?Y aaeYaacaYa).
‘ c « ©¤Y¬ aaY?Yi ¬®F®c«Y . ˜®¤aa c«ii $y=x^n+…$ c aa cY?Y
$(2)$ ? ce??aec i a®«i?® c«Ye a a ¬®© aa aeY© aaY?Yii !a?ce $x$,
?®«ac?¬ :$-2x^2n(n-1)x^{n-2}+…+4x^n+…=0$. ˜a?a c?c i a«i ?®iaeae?ae?Ya
?a? aa aeY© aaY?Y? $x$, ?®«ac?¬: $-2n(n-1)+4=0; n^2-n-2=0$. ?aai¤ $n_1=2$;
?®aYi $n_2=-1$ Y F®¤Y (aaY?Yi ¬®F®c«Y -aeY«®Y ?®«®¦?aY«i®Y c?a«®). ?a ?,
¬®F®c«Y ¬®¦Ya !eai a®«i?® ca®a®© aaY?Y?. ?eY¬ YF® c c?¤Y $y=x^2+ax+b$.
˜®¤aa c«ii c aa cY?Y $(2)$, ?®«ac?¬ $(4a+4)x+2+2a+4b=0$. ‘«Y¤®c aY«i®,
$4a+4=0, 2+2a+4b=0$. ?aai¤ $a=-1, b=0$. ?a ?, ¬®F®c«Y $y=x^2-x$ ic«iYaai
c aae¬ aYeY?Y¬.
\end{enumerate}
\subsection{?a YceY § ¤ c?.}
\begin{enumerate}
\item
„«i ®aea? ?i aYeY?i ?a Yc®© § ¤ c?
$$
a_0(x)y”+a_1(x)y’+a_2(x)y=f(x), x_0\le x\le x_1, \eqno{(1)}
$$
$$
\alpha y'(x_0)+\beta y(x_0)=0, \gamma y'(x_1)+\delta y(x_1)=0 \eqno{(2)}
$$
¤® ?®¤aa c?ai ®!eYY aYeY?Y aa cY?i $(1)$ c Fa YceY aa«®c?i $(2)$ ?
?§ ia?a aa«®c?© ®?aY¤Y«?ai (Ya«? ia® c®§¬®¦®) § cY?i ?a®?§c®«iea
?®aa®iea, ca®¤ie?a c ae®a¬a«a ®!eYF® aYeY?i. ‚ ®a«?c?Y ®a § ¤ c? a
c «ie¬? aa«®c?i¬? (§ ¤ c? ?®e?), ?a Yc i § ¤ c Y caYF¤ ?¬YYa aYeY?Y.
\item
”a?ae?Y© ?a? ?a Yc®© § ¤ c? $(1), (2)$ §ec Yaai aea?ae?i $G(x,s)$,
®?aY¤Y«Y i ?a? $x_0\le x\le x_1, x_0\le s\le x_1$, ? ?a? ? ¦¤®¬ ae??a?a®c ®¬
$s$ ?§ ?aYac « $(x_0,x_1)$ ®!« ¤ ie i ac®©aac ¬? (? ? aea?ae?i ®a $x$):
\begin{enumerate}
\item ?a? $x\ne s$ ® a¤®c«Yac®aiYa aa cY?i
$$
a_0(x)y”+a_1(x)y’+a_2(x)y=0; \eqno{(3)}
$$
\item ?a? $x=x_0$ ? $x=x_1$ ® a¤®c«Yac®aiYa § ¤ e¬ ?a Yce¬ aa«®c?i¬ $(2)$;
\item ?a? $x=s$ ® Y?aYaec ?® $x$, YY ?a®?§c®¤ i ?® $x$ ?¬YYa a? c®?,
a ce© $\frac{1}{a_0(x)}$, a.Y.
$$
G(s+0,s)=S(s-0,s), G_x’|_{x=s+0}=G_x’|_{x=s-0}+\frac{1}{a_0(x)}. \eqno{(4)}
$$
\end{enumerate}
—a®!e ©a? aea?ae?i ?a? ?a Yc®© § ¤ c? $(1), (2)$, ¤® ©a? ¤c aYeY?i
$y_1(x)$ ? $y_2(x)$ (®a«?ceY ®a $y(x)\equiv 0$) aa cY?i $(3)$, a¤®c«Yac®aiie?Y
a®®acYaaacY® ?Yac®¬a ? ca®a®¬a ?§ Fa ?cea aa«®c?© $(2)$. …a«? aea?ae?i
$y_1(x)$ Y a¤®c«Yac®aiYa aa §a ¤ca¬ ?a Yce¬ aa«®c?i¬, a® aea?ae?i ?a?
aaeYaacaYa ? YY ¬®¦® ?a? ai c c?¤Y
$$
G(x,s)=\left\{
\begin{array}{l}
ay_1(x), (x_0\le x\le s),\\
by_2(x), (s\le x\le x_1).
\end{array}
\right. \eqno{(5)}
$$
”a?ae?? $a$ ? $b$ § c?aia ®a $s$ ? ®?aY¤Y«iiaai ?§ aaY!®c ?i, ca®!e
aea?ae?i $(5)$ a¤®c«Yac®ai« aa«®c?i¬ $(4)$, a.Y.
$$
by_2(s)=ay_1(s), by_2′(s)=ay_1′(s)+\frac{1}{a_0(s)}.
$$
\item …a«? aea?ae?i ?a? $G(x,s)$ aaeYaacaYa, a® aYeY?Y ?a Yc®© § ¤ c?
$(1), (2)$ cea ¦ Yaai ae®a¬a«®©
$$
y(x)=\int\limits_{x_0}^{x_1}{G(x,s)f(s)ds.}
$$
\item ‘®!aacYe¬ § cY?Y¬ § ¤ c?
$$
a_0(x)y”+a_1(x)y’+a_2(x)y=\lambda y, \eqno{(6)}
$$
$$
\alpha y'(x_0)+\beta y(x_0)=0, \gamma y'(x_1)+\delta y(x_1)=0 \eqno{(7)}
$$
§ec Yaai a ?®Y c?a«® $\lambda$, ?a? ?®a®a®¬ aa cY?Y $(6)$ ?¬YYa aYeY?Y
$y(x)\not\equiv 0$, a¤®c«Yac®aiieYY ?a Yce¬ aa«®c?i¬ $(7)$. ˜a® aYeY?Y $y(x)$
§ec Yaai a®!aacY®© aea?ae?Y©.
\end{enumerate}
Нашли опечатку? Выделите и нажмите CTRL+Enter
