# Initial value problem and laplace transform

1. Jan 22, 2012

### azserendipity

1. The problem statement, all variables and given/known data
I understand how to do initial value problems but I'm slightly stuck when the initial values are y(0) = y'(0)=0

The question is Solve:

y''+3y''+2y=f(t), y(0)=y'(0)=0 where f(t) is a square wave.

2. Relevant equations

$\Im${y'} =s$\Im${y}-y(0)
$\Im${y''}=s$^{2}$$\Im$-sy'(0)-y'(0)

3. The attempt at a solution

I've gotten so far:

(s$^{2}$$\Im$(y)-sy(0)-y'(0))+3(s$\Im$(y)-y(0))+2$\Im$(y)=F(t)

$\Rightarrow$ (s$^{2}$-y'(0)-0)+3s-y'(0)+2=F(t)

Its then when I substitute in the initial condition I get

s$^{2}$+3s+2=F(t)

I'm not sure this is right because I cant then do partial fractions or the inverse of it to get the final answer.

The other thing is I dont understand how F(t) being a square wave affects it.

Any help would be greatly appreciated!
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

2. Jan 22, 2012

### vela

Staff Emeritus
To make things easier to type, I'm going to use the usual convention of using a capital letter to denote the Laplace transform and a lower-case letter to denote the corresponding function of time, e.g. F(s) is the Laplace transform of f(t).

You have to take the Laplace transform of both sides of the equation, so you should get
$$[s^2Y(s) - s y(0) - y'(0)] + 3[sY(s)-y(0)] + 2Y(s) = F(s)$$After substituting in the initial values, you're left with
$$(s^2+3s+2)Y(s) = F(s)$$You want to solve for Y(s) and transform back to the time domain to find y(t). In your attempt, you mysteriously dropped the Y(s).
You need to find the Laplace transform of a square wave to be able to find Y(s).

3. Jan 22, 2012

### azserendipity

I'm guessing to work out Y(s) you need to move it to the other side of the = sign so its:

(s2+3s+2)=Y(s)?

Then to find the square wave do you then need to work it out? (if so how?) or is it just something you can find out from the internet?

4. Jan 22, 2012

### vela

Staff Emeritus
That doesn't make any sense. You still need to follow the rules of algebra.

I suggest you consult your textbook.

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