Differential Equation - Nonhomogenous Hermite's Polynomial

[cse63146]

Homework Statement

Find the general solution up to degree 6 of y'' + ty' + y = e^-2t

Homework Equations

The Attempt at a Solution

I know how to solve it for y'' + ty' + y = 0, but what do I do about the e^-2t?

 

[Mark44]

Are you doing the homogeneous equation by a power series method? If so, combine your series for y'', ty', and y on the one side and equate them to the series for e^-2t.

 

[cse63146]

am I supposed to use taylor's polynomials for the e^-2t after?

 

[Mark44]

That's what I would do for starters. After all, you're only concerned with terms up to degree 6.

Start with the assumption that y = a₀ + a₁t + a₂t² and so on up to degree 8 or so (to get degree 6 term in your second derivative. Calculate y' and y'' and multiply y' by t, then add them together and group them by like powers of t. On the right side, you'll have the power series for e^-2t, which looks like 1 - 2t + (2t)²/2! + a few more terms to get to degree 6. Then equate both sides and solve for your coefficients.

 

[cse63146]

So would each group of t be equal to degree 6 of e^-2t power series.

For example

(...)t = degree 6 of e^-2t
(...)t² = degree 6 of e^-2t
(...)t³ = degree 6 of e^-2t

and so forth

or would it look like this:

(...)t = degree 1 of e^-2t
(...)t² = degree 2 of e^-2t
(...)t³ = degree 3 of e^-2t

and so forth

 

[Mark44]

This one.

(...)t = degree 1 of e-2t
(...)t2 = degree 2 of e-2t
(...)t3 = degree 3 of e-2t
and so forth

Don't forget the degree 0 term on each side.