I've been given a linear first order differential equation which can be written as $ x'(t)+p(t) x(t)= q(t)$. The general solution formula given is: $x(t)={e^{-P(t)}} \int {e^{P(t)}} \cdot q(t) dt + C {e^{-P(t)}}$ where C is our constant. Specifically, I've been given the DE $ x'(t)+2tx(t)=2t^2 $ and I've been asked if it is possible to use the general solution formula (stated above) to find a solution to the DE which only contains "regular/ordinary" functions. I know I get an erfi function, but the point of the question is to ask if it is possible to write the solution in a different way. Is this possible? If it is, how would I go about it?
So far I've tried to figure out if I might be able to use the derivative of the erfi solution to express the solution using regular functions.
If you want to avoid special functions, series expansion could be the only way.
Consider the equation $$x'+2t x=2t^2$$ and use $$x=\sum_{n=0}^\infty a_n t^n$$ This would give $$\sum_{n=0}^\infty n a_n t^{n-1}+2\sum_{n=0}^\infty a_n t^{n+1}=2t^2$$ Comparing powers, this would give $a_1=0$ and $a_2=-a_0$. Now, for a degree $m > 2$ $$(m+1)\,a_{m+1}\,t^m+2\,a_{m-1}\,t^m=0\implies a_{m+1}=-\frac{2\, a_{m-1} }{m+1 }$$