Hallo,
y'' + 2y' + 1=x*e^x
y'' + 2y' =x*e^x -1
λ^2 +2λ=0
λ1=0
λ2= - 2
yh=C1 e^(-2x) +C2
yp1=A x
yp2=Be^x +Ce^x*x
y=yp1+yp2
-->yp= Ax +B e^x +Ce^x *x
yp'= A+ Be^x +Ce^x(x+1)
yp''= Be^x +Ce^x(x+2)
------->yp ,yp',yp'' in die DGL einsetzen:
y'' + 2y' =x*e^x -1
Be^x +Ce^x(x+2) +2(A+ Be^x +Ce^x(x+1))= x*e^x -1
und vereinfachen:
3C x e^x+4C e^x +3Be^x +2A= x e^x -1
------>Koeffizientenvergleich:
x^0: 2A=-1
e^x: 4C+3B=0
x e^x: 3C=1
--->
A=-1/2
B=-4/9
C=1/3
yp= (-1/2)x -(4/9) e^x + (1/3)e^x *x
y=yh+yp
y=yh=C1 e^(-2x) +C2 +(-1/2)x -(4/9) e^x + (1/3)e^x *x