An easy question hopefully. I have the solution to a differential equation

y'+2ty=g(t)

in the form

y=\int^t_{t_{0}}g(\tau)G(t,\tau)d\tau

where G(t,\tau) is greens function thus

y=\int^t_{t_{0}}g(\tau)e^{\tau^2-t^2}d\tau

if I want to solve it for

g(t) &=t, \ 1 \leq t \leq 2

would I get

y=\int^2_{1} \tau e^{\tau^2-4}d\tau

I am just not sure about replacing the t in the equation with the upper limit.

Thanks guys/girls