Integro-Differential Equation IVP

**silencecloak** · May 13th 2010, 12:36 PM

I need some help getting started on this problem. $y^\prime - \int\limits_{0}^{t}cos(t-\tau)y(\tau) d\tau = 1, y(0) = 2$

This problem scream Convolution to me, but the $y(\tau)$ is throwing me off.

I have considered recursive integration, but am still not sure what to do with that term.

Any help? Thanks!

**TwistedOne151** · May 13th 2010, 02:41 PM

Considering the form of the convolution, consider taking the Laplace transform of the equation.

--Kevin C.

**Random Variable** · May 13th 2010, 03:08 PM

Using TwistedOne151's advice,

let $Y(s) = \mathcal{L} [y(t)]$

then $sY(s) - y(0) - \mathcal{L} [\cos t ] \mathcal{L} [y(t)] = \frac{1}{s}$

$sY(s) -2 - \frac{s}{s^{2}+1}Y(s) = \frac{1}{s}$

$Y(s)\Big(\frac{s^{3}}{s^{2}+1} \Big) = \frac{1}{s} + 2$

$Y(s) = \frac{s^{2}+1}{s^{4}} + 2 \ \frac{s^{2}+1}{s^{3}} = \frac{1}{s^{2}} + \frac{1}{s^{4}} + \frac{2}{s} + \frac{2}{s^{3}}$

then $y(t) = t + \frac{t^{3}}{6} + 2 + t^{2}$

**silencecloak** · May 13th 2010, 07:52 PM

Thanks that was a lot easier than I thought. Makes perfect sense.

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