举例说明：(x-2)*dy/dx=y 2*(x-2)^3

解：

∵(x-2)*dy/dx=y 2*(x-2)³

(x-2)dy=[y 2*(x-2)³]dx

(x-2)dy-ydx=2*(x-2)³dx

[(x-2)dy-ydx]/(x-2)²=2*(x-2)dx

d[y/(x-2)]=d[(x-2)²]

y/(x-2)=(x-2)² C (C是积分常数)

y=(x-2)³ C(x-2)

∴原方程的通解是y=(x-2)³ C(x-2)(C是积分常数

形如y'+P(X)y=Q(x)

则有通解y=e^(-∫p(x)dx)(∫Q(x)e^(∫p(x)dx)dx+C)

这里P(X)=-X,Q(X)=2X

带入得y=e^(∫xdx)(∫2xe^(-∫xdx)+C)

=e^(x^2/2)(∫2xe^(-x^2/2)dx+C)

=e^(x^2/2)(-2∫e^(-x^2/2)d(-x^2/2)+C)

=e^(x^2/2)(-2e^(-x^2/2)+C)

=-2+Ce^(x^2/2)其中C是任意常数