Question relating to solving equations with Green's functions

Feb 2011
An easy question hopefully. I have the solution to a differential equation

\(\displaystyle y'+2ty=g(t)\)

in the form

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

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

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

if I want to solve it for

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

would I get

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

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

Thanks guys/girls