Differential Equations: Initial Value Problem

Question

$y = -t\cos t - t $ is a solution of the initial value problem: $$t\frac{dy}{dt} = y + t^2\sin t$$ where $$y(\pi) = 0$$

I know how to find out if $y$ is a solution, but I'm not sure how to do it when $y(\pi) = 0$.

Answer 1

If you have verified that the given equation is a solution to the differential equation, it just remains to check that $y(\pi)=0$.

$$ y(\pi)=-\pi\cos(\pi)-\pi=\pi-\pi=0 $$ Indeed, the solution given satisfies the boundary conditions and is solution to the IVP.

Answer 2

The $y(\pi)$ shouldn't matter if you know already how to do it without the initial value. For the given y we see that $y(\pi)$ is equal to 0 so now just plug in $\frac{\text{d}y}{\text{d}t}$ and prove that it works out

Answer 3

$$y(\pi)=-\pi\cos(\pi)-\pi=\pi-\pi=0$$ is the required solution