Need help with a differential equation!

Question

Verify that y=-tcos(t)-t is a solution of the initial-value problem, t(dy/dt)=y+t²sin(t), y(pi)=0

Answer 1

First find the derivative of the solution.

dy/dt = -cos(t) + tsin(t) - 1

Now plug everything in and confirm that this is a solution. Also confirm the initial value.

t[-cos(t) + tsin(t) - 1] = -tcos(t) - t + t²sin(t)

Using distribution on the left side, we get -tcos(t) + t²sin(t) - t. This equals the right side, so first part is done.

y(π) = -πcos(π) - π = -π(-1) - π = π - π = 0

Since both parts check out, this is a solution to this differential equation.

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