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

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.

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 $$y(t_0)=e^{-t_0^3/3} \left(0 + c_2\right)\implies c_2=e^{t_0^3/3}y(t_0).$$
• Thanks. You're right, I forgot the parentheses. Can you please write down more steps of how you got from the first equation to the other? Also what confusion would it make if i kept the variable $t$? Do you have any example? – Lauren Sin Apr 16 '20 at 11:49
• See my edit. The confusion is that we use the same symbol $t$ for two different things: the upper limit and the variable of integration. If we are aware of such a distinction we may also write $$y(t)=e^{-t^3/3} \left(\int_{t_0}^t e^{t^3/3} {dt} + c_2\right).$$ The variable of integration is a dummy variable: mathworld.wolfram.com/DummyVariable.html – Robert Z Apr 16 '20 at 12:03