How to Solve a Laplace Transform Problem for y'' + 4y = 8

[pakmingki2]

Homework Statement

find solution using laplace transforms


y'' + 4y = 8



alright, so i did the laplace transform of both sides and i get

(s^2 + 4)L(y) - 11s - 5 = 8/s

so i isolate L(y) and i get this expression:

L(y) = (11s^2 + 5s + 8)/(s*(s^2 + 4))

however, the textbook says the answer is:

L(y) = 2/s + (9s + 5)/(s^2 + 4)

And i don't know how to get from my expression to the book's.
I'm pretty good at doing inverse laplace transforms, its just that i can't seem to do the algebra right.

CAn someone help me see how to get to the right expression for L(y)?
thanks!

 

[benorin]

Use partial fraction decomposition:

Start off by setting

\frac{11s^2 + 5s + 8}{s(s^2 + 4)} = \frac{A}{s}+\frac{ Bs + C}{s^2 + 4}

multiply through by the common demoninator s(s^2 + 4) and plug-in 3 different values of s to generate 3 equations involving A,B, \mbox{ and }C. Nice values of s here include s=0 (this will give the value of A), and either s=\pm 1 (which gives 2 equations in A\mbox{ and }B) or s=2i (which gives the values of A\mbox{ and }B by equating real and imaginary parts). Enjoy :).

Dear Moderator: I know this post goes beyond what we by rule give in guidance in solving a problem, yet I offer this apology: I could not quickly find a web page that gave instructions for the above easy method of partial fraction decomposition to my satisfaction: hence my post.

 

[pakmingki2]

well don't worry i found out a way anyways.

so i got:

L(y) = 8/s(s^2 + 4) + (11s + 5)/(s^2 + 4)

i won't type it all out cause it's annoying, but what i did was i partial fraction decomposed 8/s(s^2 + 4) and i expanded all the terms and equated coeficcients cause i hate dealing with complex numbers when using partial fractions. So i find the values of A B C and combine all the terms in it comes out to the expression i was looking for.