dy+ P(x)y = Q(x)dxy= C(x)e / P(n)adx求导,得对y'= C(x)e J (m)dr- P()dx.(-P(x)+C(x)edy将y和y代入原方程得+ P(x)y = Q(x)dxC(x)e -/ P(x)dr+C()[-P(x)]e f p()dr+ P(x)C(x)e + P()g)=Q(x)从而C(x)满足方程 C(x)e-[ P(r)de× = Q(x)
+ = − P x x y C x e ( )d ( ) 将y和y代入原方程, Q(x) + − − P x x − P x x C x e C x P x e ( )d ( )d ( ) ( ) ( ) = + − P x x P x C x e ( )d ( ) ( ) 从而C(x)满足方程 ( ) , ( )d 对 = 求导 − P x x y C x e 得 C(x) [−P(x)] − P x x e ( )d 得 ( ) ( ) d d P x y Q x x y + = ( ) ( ) ( )d C x e Q x P x x = − ( ) ( ) d d P x y Q x x y + =
P(x)dx即fc'(x)dx=[e(x)edxC(m)=Je(x)e ax)ardx +C-J P(x)dxdy设y=是+P(x)y =Q(x)的解dx一阶线性非齐次微分方程的通解为 = (waf (xel ()d x+ 常数变易法把齐次方程通解中的常数变易为待定函数的方法
即 = P x x C x Q x e ( )d ( ) ( ) C x Q x e x P x x ( ) ( ) d ( )d = +C 一阶线性非齐次微分方程的通解为 [ ( ) d ] ( )d ( )d y e Q x e x C P x x P x x + = − = − P x x y C x e ( )d 设 ( ) 常数变易法 把齐次方程通解中的常数变易为 待定函数的方法. dx dx ( ) ( ) . d d 是 P x y Q x 的解 x y + =
dy+ P(x)y = Q(x)dx一阶线性方程解的结构P(x)dxJ=e p(x)dr,tfe(x)e)dx+CJCe- / p(n)ax-J P(x)dxP(x)dx.Je(x)eJdx+对应齐次非齐次方程的一个特解方程通解注一阶线性方程解的结构及解非齐次方程的常数变易法对高阶线性方程也适用
+ = − P x x Ce ( )d 对应齐次 非齐次方程的一个特解 方程通解 [ ( ) d ] ( )d ( )d y e Q x e x C P x x P x x + = − 一阶线性方程解的结构 e Q x e x P x x P x x ( ) d ( )d ( )d − 注 一阶线性方程解的结构及解非齐次方程 的常数变易法对高阶线性方程也适用. ( ) ( ) d d P x y Q x x y + =
P(x)dxy=eJP(x)dr1[Q(x)edx + Clsin x的通解例 求方程+V=xxsinx1阶线性非一解 P(x)=00齐次方程xx2dxdxsinxxxedx +CeV=x1sin xdx + C)= =(- cos x + C)X
. 1 sin 求方程 的通解 x x y x y + = , 1 ( ) x P x = , sin ( ) x x Q x = = − x x y e d 1 ( ) = x x + C x sin d 1 ( x C ) x = − cos + 1 解例 一阶线性非 齐次方程 x sin x x x e d 1 dx +C [ ( ) d ] ( )d ( )d y e Q x e x C P x x P x x + = −