# Homework Help: Using Stepfunction to solve IVP

1. Oct 25, 2009

### Susanne217

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

I am given the task for finding the laplace transformation given

$$y'' + 4y = 8t^2$$ if 0<t<5 and 0 if t>5 y(1) = 1+cos(2) $$y'(1)= 4-2sin(2)$$

2. Relevant equations

3. The attempt at a solution

I know that the above problem can be written $$y''+4y=8 \cdot u(t)$$

where u(t) is the step function.

But what is my next step?

$$\mathcal{L}(y''+4y) = \mathcal{L}(8t^2)$$

which equals

$$\mathcal{L}(y'') + \mathcal{L}(4y) = 8\mathcal{L}(t^2)$$

$$\mathcal{L}(y'') = s \cdot \mathcal{L}(y') - y'(1)$$

$$\mathcal{L}(y') = s \cdot \mathcal{L}(y) - y(1)$$

not sure what to do about the left side by the rightside that is $$Y(s^2+ 4) = \frac{16}{s^3} + s(1+cos(2)) + (4-2sin(2))$$

then $$Q(s)= \frac{1}{s^2+4}$$

then $$Y(s) = s(1+cos(2)) + 4-2sin(2)) \cdot \frac{1}{s^2+4} + \frac{16}{s^3} \cdot \frac{1}{s^2+4}$$ Is this true or am I totally fubar?

Last edited: Oct 25, 2009