# Thread: Integro-Differential Equation IVP

1. ## Integro-Differential Equation IVP

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

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

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

Any help? Thanks!

2. ## Laplace

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

--Kevin C.

3. Using TwistedOne151's advice,

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

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

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

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

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

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

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