# ODE by Laplace transform

1. Jan 12, 2016

### roughwinds

• Member warned about posting without the homework template
1. The problem statement, all variables and given/known data
Use Laplace transform to solve the following ODE

2. Relevant equations
xy'' + y' + 4xy = 0, y(0) = 3, y'(0) = 0

3. The attempt at a solution
$L(xy'') = -\frac{dL(y'')}{ds}$

$L(4xy) = -\frac{4dL(y)}{ds}$

$L(y'') = s²L(y) - sy(0) - y'(0) = s²L(y) -3s$

$L(y') = sL(y) - sy(0) - y(0) = sL(y) - 3$

$-\frac{d(s²L(y)-3s)}{ds} + sL(y)-\frac{4dL(y)}{ds} =0$

$-\frac{d(s²L(y))}{ds} + 3 + sL(y) - 3 -\frac{4dL(y)}{ds} =0$

$-\frac{d(s²L(y))}{ds} + sL(y) -\frac{4dL(y)}{ds} =0$

$-\frac{dL(y)(s²+4)}{ds} + sL(y) =0$ (1)

$\frac{dL(y)}{L(y)} = \frac{sds}{s²+4}$

Integrating both sides

$ln(L(y)) = \frac{ln(s²+4)}{2} + c$

$L(y) = c\sqrt{s²+4}$

I realized that if at (1) I use $\frac{L(y)(s² + 4)}{ds} + sL(y) =0$ instead I'll reach the right answer according to wolfram, but I can't figure out what I'm doing wrong to end up with that negative sign.

http://www.wolframalpha.com/input/?i=xy''+++y'+++4xy+=+0,+y(0)+=+3,+y'(0)+=+0

Last edited: Jan 12, 2016
2. Jan 12, 2016

### Samy_A

Something went wrong from:
$-\frac{d(s²L(y))}{ds} + 3 + sL(y) - 3 -\frac{4dL(y)}{ds} =0$
to
$-\frac{s²L(y)}{ds} + sL(y) -\frac{4dL(y)}{ds} =0$

Check what you did with the $-\frac{d(s²L(y))}{ds}$ term.

3. Jan 12, 2016

### roughwinds

It's supposed to be
$-\frac{d(s²L(y))}{ds} + sL(y) -\frac{4dL(y)}{ds} =0$
fixed it on the original post.

4. Jan 12, 2016

### Samy_A

Ok, so now continue your derivation from that.

5. Jan 12, 2016

### roughwinds

$-\frac{d(s²L(y))}{ds} = -2sL(y) - \frac{s²d(L(y))}{ds}$
$-2sL(y) - \frac{s²d(L(y))}{ds} + sL(y) -\frac{4dL(y)}{ds} =0$
$- \frac{s²d(L(y))}{ds} - sL(y) -\frac{4dL(y)}{ds} =0$
$\frac{s²d(L(y))}{ds} + sL(y) +\frac{4dL(y)}{ds} =0$
$\frac{d(L(y))(s²+4)}{ds} + sL(y) =0$

Thanks, I initially solved as if s² wasn't part of the derivative.