Integro-Differential Equation IVP

May 2008
138
0
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)
 
Dec 2007
276
143
Anchorage, AK
Laplace

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

--Kevin C.
 
May 2009
959
362
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 \(\displaystyle y(t) = t + \frac{t^{3}}{6} + 2 + t^{2} \)
 
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May 2008
138
0
Thanks that was a lot easier than I thought. Makes perfect sense.