Initial value problem and laplace transform

[azserendipity]

Homework Statement

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.

Homework Equations

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

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 can't then do partial fractions or the inverse of it to get the final answer.

The other thing is I don't understand how F(t) being a square wave affects it.


Any help would be greatly appreciated!

 

[vela]

Homework Statement

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.

Homework Equations

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

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)

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).

$\Rightarrow$ (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 can't then do partial fractions or the inverse of it to get the final answer.

The other thing is I don't understand how F(t) being a square wave affects it.


Any help would be greatly appreciated!

You need to find the Laplace transform of a square wave to be able to find Y(s).

 

[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?

 

[vela]

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)?

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

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?

I suggest you consult your textbook.