When and why do I change the variable when solving DE using definite integrals?

Question

I've seen similar questions here but none of them answered my problem. Or at least not the way that I would understand. I am trying to solve differential equation $$y'+t^2y=1.$$ I used the method of integrating factor - for this case $e^{t^3/3}$. By multiplying the equation i get $$e^{t^3/3}y'+e^{t^3/3}t^2y=e^{t^3/3}.$$ Now here is when I got confused. I should now integrate both sides (using indefinite integral?) which should result in $$c_1+e^{t^3/3}y=e^{t^3/3}.$$ So the solution should be $$y=e^{-t^3/3} \int e^{t^3/3} {dt} + c_2.$$ Now when I rewrite the integral in terms of a definite integral I get $$y=e^{-t^3/3} \int_{t_0}^{t} e^{s^3/3} {ds} + c_2.$$ My questions are first one - when do I change the variable in integral - is it because I change it from indefinite to definite? If so, why do I need to change the variable at all?

Second question - how do I manage the constants? Should I just keep it as $C$ or do I need to consider the initial condition too, since I change the integral from indefinite to definite? If so, how?

Thanks.

Answer 1

Your solution should be $$y(t)=e^{-t^3/3} \left(\int e^{t^3/3} {dt} + c_2\right).$$ (pay attention to the enclosing parentheses). In terms of a definite integral, it can be written as $$y(t)=e^{-t^3/3} \left(\int_{t_0}^t e^{s^3/3} {ds} + c_2\right)$$ where we renamed the variable inside the integral just to avoid confusion with the variable at the upper limit of the integration.

We may also find $c_2$ in terms of $y(t_0)$: by letting $t=t_0$ we get $$y(t_0)=e^{-t_0^3/3} \left(0 + c_2\right)\implies c_2=e^{t_0^3/3}y(t_0).$$