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$B0lHL$N(B $f(x)$ $B$KBP$7$F!"HsF1

$$y''+p y'+q y=f(x)$$ (7)

$B$NFC2r$r5a$a$k$K$O!"(B (1) Laplace $BJQ49$rMxMQ$9$kJ}K!(B, (2) $BDj?tJQ2=K!$J$I?'!9$JJ}K!$,$"$k$,!"(B $B$3$3$G$O=i4|CMLdBj$N(B Green $B4X?t(B$B$rMQ$$$kJ}K!$r>R2p$9$k!#(B

$B$3$NDjM}$K8=$l$?4X?t(B $G$ $B$N$3$H$r(B $BHyJ,J}Dx<0(B (7) $B$N(B $B=i4|CMLdBj$N(B Green $B4X?t(B$B$H$h$V!#(B

$B>v$_9~$_$rMQ$$$k$H!"(B $B>e$N(B (9) $B$N(B $u$ $B$O(B $u=G\ast f$ $B$H=q$1$k$3$H$,J,$+$k!#(B $B>v$_9~$_$O>e$NDjM}$N>ZL@$K$b3hLv$9$k!#(B $B$=$N$?$a$K>/$7=`Hw$7$h$&!#(B

(1) $B$O4JC1$G$"$k$N$G>JN,$9$k!#(B (2), (3) $B$O1i=,LdBj$H$9$k!#(B (4) $B$O0J2<$N5DO@$KI,MW$,$J$$$N$G>JN,$9$k(B³$B!#(B $\qedsymbol$

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$$y'-a y=f(x),\quad y(0)=0$$

$B$N2r$O(B

$$y=\int_0^x e^{a(x-y)}f(y) \D y$$

$B$G$"$k$3$H$r;W$$=P$7$F$*$/!#>v$_9~$_$rMQ$$$k$H(B

$$y=(e^{a x}\ast f)(x)$$

$B$H$b=q$1$k!#(B

$BDjM}(B $A(x):=e^{\alpha x}$, $B(x):=e^{\beta x}$ $B$H$*$/!#(B $u$ $B$,(B

$$u''+p u'+q u=f(x),\quad u(0)=u'(0)=0$$

$B$rK~$?$9$H$9$k$H$-!"(B $v:=u'-\beta u$ $B$H$*$/$H!"(B

$$v'-\alpha v=(v''-\beta v')-\alpha(v'-\beta) =v''-(\alpha+\beta) v'+\alpha\beta v =v''+p v'+q v=f(x),$$

$$v(0)=u'(0)-\beta u(0)=0-\beta\cdot 0=0$$

$B$G$"$k$+$i!">e$K=q$$$?Cm0U$h$j(B

$$v(x)=(A\ast f)(x).$$

$B$H$3$m$G(B

$$u'-\beta u=v(x),\quad u(0)=0$$

$B$G$"$k$+$i!"(B$u=B\ast v$. $B$f$($K(B

$$u=B\ast v=B\ast(A\ast f)=(B\ast A)\ast f.$$

$B$f$($K(B $G:=B\ast A$ $B$H$*$/$H!"(B$u=G\ast f$ $B$H$J$k!#(B $B0J2<(B $G$ $B$r6qBNE*$K7W;;$7$F5a$a$h$&!#(B

$\alpha\ne \beta$ $B$N>l9g$O(B

$$G(x) = \int_0^x B(x-y)A(y) \D y =\int_0^x e^{\beta(x-y)}e^{\alpha y} \D y =e^{\beta x}\int_0^x e^{(\alpha-\beta) y} \D y$$

$$=e^{\beta x} \left[\frac{e^{(\alpha-\beta) y}}{\alpha-\beta}\right]_0^x = \dfrac{e^{\alpha x}-e^{\beta x}}{\alpha-\beta}.$$

$B0lJ}!"(B $\alpha=\beta$ $B$N>l9g$O!"(B

$$G(x) = \int_0^x B(x-y)A(y) \D y =\int_0^x e^{\alpha(x-y)}e^{\alpha y} \D y =e^{\alpha x}\int_0^x \D y =x e^{\alpha x}. \qed$$

ARRAY(0xfd7e1c)