求微分方程的通解

设x=e^t
则d^2 y / dt^2 - 5dy / dt + 6y = e^t
y = C1 * e^(3t) + C2 * e^(2t) + 1/2 e^t
=C1 * x^3 + C2 * x^2 + x/2设x=e^t则d^2 y / dt^2 - 5dy / dt + 6y = e^t这个怎么出来的啊dx/dt = xdy/dx = dy/(xdt)=(dy/dt)/xd^2 y / d x^2 = d((dy/dt)/x)/dx = d((dy/dt)/x)/(xdt) = (d^2 y / dt^2) / x^2 -(dy/dt)/x^2因此x dy/dx = dy/dtx^2 d^2 y / dx^2 = d^2 y / dt^2 - dy/dtd^2 y / dt^2 - 5dy / dt + 6y = e^ty = C1 * e^(3t) + C2 * e^(2t) + 1/2 e^t中1/2 e^t怎么出来的啊可设特解= k e^t代入方程, 可得k-5k+6k=1=> k = 1/2因此, 特解等于 1/2 e^t