Differential Equations: Non-homogeneous Series Expansion

[Bryon]

Homework Statement

y'' + y' + y = 1 + x + x²

Homework Equations

y = Ʃ C_N*x^N N starts at 0
y' = Ʃ N*C_N*x^(N-1) N starts at 1
y'' = Ʃ N*(N-1)*C_N*x^(N-2) N starts at 2

3. The Attempt at a Solution [/]
I know how solve the equations using series when the equation would equal to 0. My main question about using series on a non-homogeneous differential equation is whether or not the varialbes on the right side have the C_x coefficients? Or would they be paired up with the x, x², etc? I think I need some quick clarification on this.

Thanks!

 

[LCKurtz]

Homework Statement

y'' + y' + y = 1 + x + x²

Homework Equations

y = Ʃ C_N*x^N N starts at 0
y' = Ʃ N*C_N*x^(N-1) N starts at 1
y'' = Ʃ N*(N-1)*C_N*x^(N-2) N starts at 2

3. The Attempt at a Solution [/]
I know how solve the equations using series when the equation would equal to 0. My main question about using series on a non-homogeneous differential equation is whether or not the varialbes on the right side have the C_x coefficients? Or would they be paired up with the x, x², etc? I think I need some quick clarification on this.

Thanks!

The C_n's only appear in your expressions for y and its derivatives. But you must take the powers of x on the other side into account for your recursion formulas. I assume you know that series isn't the easiest way for this problem.

 

[Bryon]

Thanks for clearing that up. The instructor covered only homogenous problems, and when I ran into one of these I was not entirely sure how to solve it with series.