Answer to Question #155975 in Differential Equations for Mike

Question

y''+y'-2y=e^x+4sinx+x^2-x operator method differential equations

Answer 1

y'=Dy

y"=D²y

(D²+D-2)y=e^x+4sin(x)+x²-x

Common solution

D²+D-2=0

$D1=(-1+ \sqrt{1+8})/2=1$

$D2=(-1- \sqrt{1+8})/2=-2$

y(x)=c₁e^x+c₂e^-2x

Particular solution for e^x

De^kx=ke^kx

D²e^kx=k²e^kx

k=1

De^x=e^x

D²e^x=e^x

y(x)=e^x+e^x-2=2e^x-2

Particular solution for sin(x)

Dsin(kx)=kcos(kx)

D²sin(kx)=-k²sin(kx)

k=1

D4sin(x)=4cos(x)

D²4sin(x)=-16sin(x)

y(x)=-16sin(x)+4cos(x)-2

Particular solution for x²

Dx^k=kx^k-1

D²x^k=k(k-1)x^k-2

k=2

Dx²=2x

D²x²=2(2-1)x^2-2=2

y(x)=2+2x-2=2x

Particular solution for x

k=1

Dx=1

D²x=0

y(x)=1-2=-1

Answer: y(x)=c₁e^x+c₂e^-2x +2e^x-2-16sin(x)+4cos(x)-2+2x-1=c₁e^x+c₂e-^2x+2e^x-16sin(x)+4cos(x)+2x-5