Question

微分方程的问题

请问下面的题，f(x)的条件是连续 可以两边直接求导么

Answer 1

let
u=x-t
du=-dt
t=0, u=x
t=x, u=0
∫(0->x) f(x-t) dt
=∫(x->0) f(u) (-du)
=∫(0->x) f(t) dt
f(x)-2∫(0->x) f(x-t) dt = e^x
f(x)-2∫(0->x) f(t) dt = e^x
两边求导
f'(x) - 2f(x) =e^x
let
yp= Ae^(2x)
yg= Be^x
yg' -2yg= e^x
-Be^x e^x
B=-1
f'(x) - 2f(x) =e^x
f(x) = Ae^(2x) -e^x
f(0) =1
A -1=1
A=2
ie
f(x) = 2e^(2x) -e^x

Answer 2