# Integro-Differential Equation IVP

#### silencecloak

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!(Happy)

#### TwistedOne151

Laplace

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

--Kevin C.

#### Random Variable

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

• silencecloak

#### silencecloak

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