Question

So derive (by hand) the closed-form solutions to the following three problems using this approach: dy (a)| +5y= 6 cos(t) sin(t); y(0)=5 (b) | dy-y=cos2(t): y(0)=1 dy sin 0 (c)-+ 3y ;y(0) = 2 dt 2S1

Answer 1

Dy/dt + 5y = 6 cos (t) sin (t) use sin (2t) = 2 sin (t) cos (t) dy/dt + 5y = 3 sin (2t) - (1) first order linear differential equation find the integrating factor If = e^integral 5 dt = e^5t multiplying by If in (1) e^5t (dy.dt) + 5 e^5t y = 3 e^5t sin (2t) d/dt (e^5t y) = 3 e^5t sin (2t) dt integral d (e^5t y) = integral 3 e65t sin (2 t) dt e^5t y = 3 integral e^5t sin (2t) solve this by applying integral by parts \r\n (e^5t y) = 3 integral e^5t sin (2t) dt - (ii) integral e^5t sin (2t) dt = e^5t sin (2t)/5 - (2 e^5t cos (2t)/25 - integral - 4 e^5t cos (2t)/24 = e^5tr sin (2t)/5 - (2 e^5t cos (2t)/25 + 4/25 integral e^5t sin (2t) dt The integral e^5t sin (2t) appears again on the right side of the equation we can ve solve for it.