Laplace tranform problem

1. Nov 25, 2007

pakmingki2

1. The problem statement, all variables and given/known data
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 dont know how to get from my expression to the book's.
I'm pretty good at doing inverse laplace transforms, its just that i cant seem to do the algebra right.

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

2. Nov 26, 2007

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.

3. Nov 26, 2007

pakmingki2

well dont worry i found out a way anyways.

so i got:

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

i wont 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.
The text contains a mathematical problem related to solving a differential equation using Laplace transforms, and it includes a detailed question and a solution approach provided by another user. The original poster also describes how they solved the problem.
single_question_with_answer
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
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- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
Laplace tranform problem

1. Nov 25, 2007

pakmingki2

1. The problem statement, all variables and given/known data
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 dont know how to get from my expression to the book's.
I'm pretty good at doing inverse laplace transforms, its just that i cant seem to do the algebra right.

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

2. Nov 26, 2007

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.

3. Nov 26, 2007

pakmingki2

well dont worry i found out a way anyways.

so i got:

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

i wont 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.
user
The text contains a mathematical problem related to solving a differential equation using Laplace transforms, and it includes a detailed question and a solution approach provided by another user. The original poster also describes how they solved the problem.

TYPE: SINGLE_QUESTION_WITH_ANSWER