Particular Solution of ODE using Annihilator

[trust]

Homework Statement

By using the method of differential operators, solve

y''+2y'+2y=2e^-xsinx

1. Determine what is the annihilator of the inhomogeneous term.

2. Find a particular solution.

3. Write the general solution for the equation.

Homework Equations

xⁿe^axsin(bx) --> annihilated by --> (D²-2aD+(a2+b2))ⁿ⁺¹

The Attempt at a Solution

1. No problem with this. Annihilated by D²+2D+2I

2. and 3. Not sure how to get these.

I multiple the annihilator by both sides of the equation. I then get ((D+1)²+1)²=0 From here I'm not sure what to do.

Thanks

 

[vela]

The idea is that because (D²+2D+2)(e^-xsin x) = 0, it follows that

(D²+2D+2)[(D²+2D+2)y] = (D²+2D+2)[2e^-xsin x] = 0

In other words, you want to solve the homogeneous equation

(D²+2D+2)(D²+2D+2)y = 0

 

[trust]

This is how I think I should solve it from my notes:

((D+I)²+I)((D+I)²+I)y=0

Basis for kernel:

((D+I)²+I): {e^xcos(x), e^xsin(x)}
((D+I)²+I): {e^xcos(x), e^xsin(x)}

y_p= Ae^xcos(x) + Be^xsin(x)
y_H= c₁e^xcos(x) +c₂e^xsin(x)

y=y_p+y_H

y= Ae^xcos(x) + Be^xsin(x) + c₁e^xcos(x) +c₂e^xsin(x)

However, the answer (was given) for y_p is y_p=-xe^-xcosx and for the general solution. is y=e^-x(c₁cosx+c₂sinx-xcosx). I'm not sure where I am going wrong.

 

[vela]

It's a good first guess, but the problem is you have repeated roots because the annihilator is the identical to the differential operator for the original equation. What do you do when you have repeated roots?

 

[trust]

((D+I)²+I)²y=0

Basis for kernel:

((D+I)²+I)²: {e^xcos(x), e^xsin(x), xe^xcos(x), xe^xsin(x)}


yp= A(x)e^xcos(x) + B(x)e^xsin(x)
yH= c1e^xcos(x) +c2e^xsin(x)

y=yp+yH

y= A(x)e^xcos(x) + B(x)e^xsin(x) + c1e^xcos(x) +c2e^xsin(x)

Is this correct? I still think something is wrong because I'm not getting the correct answer

 

[vela]

I'm not sure if this is what you meant by what you wrote, but the particular solution is $y_p = A xe^x\cos x + B xe^x\sin x$. So now you plug this back into the original differential equation and solve for constants A and B.