- #1

roughwinds

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Member warned about posting without the homework template

## Homework Statement

Use Laplace transform to solve the following ODE

## Homework Equations

xy'' + y' + 4xy = 0, y(0) = 3, y'(0) = 0

## The Attempt at a Solution

[itex]L(xy'') = -\frac{dL(y'')}{ds}[/itex]

[itex]L(4xy) = -\frac{4dL(y)}{ds}[/itex]

[itex]L(y'') = s²L(y) - sy(0) - y'(0) = s²L(y) -3s[/itex]

[itex]L(y') = sL(y) - sy(0) - y(0) = sL(y) - 3[/itex]

[itex]-\frac{d(s²L(y)-3s)}{ds} + sL(y)-\frac{4dL(y)}{ds} =0[/itex]

[itex]-\frac{d(s²L(y))}{ds} + 3 + sL(y) - 3 -\frac{4dL(y)}{ds} =0[/itex]

[itex]-\frac{d(s²L(y))}{ds} + sL(y) -\frac{4dL(y)}{ds} =0[/itex]

[itex]-\frac{dL(y)(s²+4)}{ds} + sL(y) =0[/itex] (1)

[itex]\frac{dL(y)}{L(y)} = \frac{sds}{s²+4}[/itex]

Integrating both sides

[itex]ln(L(y)) = \frac{ln(s²+4)}{2} + c[/itex]

[itex]L(y) = c\sqrt{s²+4}[/itex]

which won't lead me to the right answer.

I realized that if at (1) I use [itex]\frac{L(y)(s² + 4)}{ds} + sL(y) =0[/itex] 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

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