f''(x)+f(x)=2e^x

Lian lide

f'(x)-3f(x)=-2e^x ②

F'(x)-3f(x)=0

Get ln|f(x)|=3x+C 1。

f(x)=Ce^(3x)=u(x)e^(3x)

Substitution of constant variation method into ②

u'(x)e^(3x)=-2e^x

Then u' (x) =-2e (-2x)

So u (x) = e (-2x)+c

Then f (x) = u (x) e (3x)

=(e^(-2x)+C)e^(3x)

=e^x+Ce^(3x)

Substitute (1)

C=0

So f (x) = e x