How Do You Solve a Complex IVP with Sinusoidal and Polynomial Terms?

[cronxeh]

Ok I'm stuck on this problem - and it didnt particularly look hard when I started doing it, oh say about a good 2 hours ago.. hmm

Solve the IVP
$$2y'' + 3y' + y = t^2 + 85 sin(2t), y(0) = -2, y'(0) = 0.$$

My homogeneous solution is $$c1y1 + c2y2$$

$$y1 = e^\frac{-t}{2}, y2 = e^-^t, Wronskian = e^\frac{-3t}{2}+e^\frac{-t}{2}$$

So for a particular solution I've tried 2 different methods

1. Undetermined Coefficients
Assumed solution was $$At^2 + Bt + C + Dcos(2t) + Esin(2t)$$

$$yp' = 2At + B - 2Dsin(2t) + 2Ecos(2t)$$
$$yp'' = 2A - 4Dcos(2t) -4Esin(2t)$$

After pluggin this back into original equation $$2y'' + 3y' + y$$ I got the following equalities:
$$At^2 + Bt + C + 6At + 3B + 4A = t^2$$
$$cos(2t)(-7D + 6F) = 0$$
$$sin(2t)(-7F - 6D) = 85sin(2t)$$

Fiddling around with the equalities I got C=0, D=-6, F=-7

$$A=\frac{3t}{3t+14}, B=\frac{-4t}{3t+14}$$

After assembling the whole thing together I don't get the right hand side after integrating to check the solutions, and so even when I derive the y=yp+yh solution to get my constants for homogeneous equation I get wrong coefficients c1 and c2 for yh - an infinite number of them, so I'm pretty sure my solution is wrong

Should I have multiplied the assumed solution by t anywhere?

2. Variation of Parameters

$$Yp = -y1 \int \frac{y2 g(t) dt}{Wronskian} + y2 \int \frac{y1 g(t) dt}{Wronskian}$$
where $$g(t) =\frac{ t^2 + 85sin(2t) } {2}$$
The result was extremely long and contained a lot of exponentials, and I assumed it to be incorrect, although any comment on the correct answer you got would be appreciated

 

[James R]

In your undetermined coefficients solution, why do A and B depend on t? A and B are supposed to be constants, right?

The particular solution I get is:

$$y = t^2 - 6t + 14 -6\cos 2t -7 \sin 2t$$

 

[saltydog]

Ok I'm stuck on this problem - and it didnt particularly look hard when I started doing it, oh say about a good 2 hours ago.. hmm

Solve the IVP
$$2y'' + 3y' + y = t^2 + 85 sin(2t), y(0) = -2, y'(0) = 0.$$

My homogeneous solution is $$c1y1 + c2y2$$

$$y1 = e^\frac{-t}{2}, y2 = e^-^t, Wronskian = e^\frac{-3t}{2}+e^\frac{-t}{2}$$

So for a particular solution I've tried 2 different methods

1. Undetermined Coefficients
Assumed solution was $$At^2 + Bt + C + Dcos(2t) + Esin(2t)$$

$$yp' = 2At + B - 2Dsin(2t) + 2Ecos(2t)$$
$$yp'' = 2A - 4Dcos(2t) -4Esin(2t)$$

I get the following equations:

4A+3B+C=0
6A+B=0
A=1
6E-7D=0
-7E-6D=85

Edit: Suppose I should compare this to the results reported by James . . .
Ok, I'm cool.

 

[cronxeh]

In your undetermined coefficients solution, why do A and B depend on t? A and B are supposed to be constants, right?

The particular solution I get is:

$$y = t^2 - 6t + 14 -6\cos 2t -7 \sin 2t$$

Doh.. of course! So since $$At^2 = t^2$$ then $$A=1$$ and I simply equate the other terms to 0. Thanks.

 

[GCT]

With variation of parameters you typically want to work out the entire step, especially if you're new to it, I prefer the practical variation of parameters approach v.s. plugging into the wronskian derivation. Reduction of order should work with this also, except try using your y1 intstead of the simpler y2. Have you gotten to the laplace transform yet?

 

[GCT]

I think that you will have more cancellations with the y1, however if you need to go with y2 (using reduction of order), you'll probably need to use euler's form of sin(2t) to solve for your first integral