# Laplace transforms

1. Mar 24, 2013

### zhaniko93

1. The problem statement, all variables and given/known data

I'm given the transfer function of LTI system is $\frac{1}{s^2 + 4}$

2. Relevant equations

H(s) = $\frac{Y(s)}{X(s)}$

3. The attempt at a solution

first of all I had to find diff. equations of the system. I found that it's y'' + 4*y = x;
Then they asked to find such initial conditions that if I applied unit impulse at input, I got y=0 for time t>=0; I took Laplace transform of diff. equation with initial conditions and got that s^2Y - sY(0) - Y'(0) + 4Y = 1; I want y=0, so Y=0 and 1+sY(0) + Y'(0) = 0; I'm wrong in something....

2. Mar 24, 2013

### zhaniko93

Matlab

Code (Text):
syms y(t) x(t) a t
Dy = diff(y);
A = dsolve(diff(y, 2) + 4*y == dirac(t), y(0) == 0, Dy(0) == -1);
ezplot(A, [0 10])

pretty(laplace(A))
I get something slightly different in matlab

3. Mar 24, 2013

### milesyoung

4. Mar 24, 2013

### zhaniko93

Thanks milesyoung, it helped!

5. Mar 24, 2013

### rude man

You should use y for the time functions and Y for the transformed variables.

You are right in saying y(0+) = 0 but what about y'(0+)?

6. Mar 24, 2013

### milesyoung

He/she wrote the correct initial conditions in the code segment.