Question

I need help solving y^(prime) + 2xy = 2x^3. It's an ordinary differential equation.

Answer 1

\[y \prime+2xy=2x^3\]
\[I.F.=e ^{\int\limits 2xdx}=e ^{x^2}\]
\[C.S. ~is~y~e ^{x^2}=\int\limits 2x^3 .e ^{x^2}dx=\int\limits~x^2~e ^{x^2}(2x)dx+c=I _{1}+c\]
\[I _{1}=\int\limits x^2~e ^{x^2}(2x)dx\]
put \[x^2=t,2x~dx=dt\]
\[I _{1}=\int\limits t~e^t~dt\]
integrate by parts and then replace the value of t.