Originally Posted by

**pickslides** You need to use the integrating factor method on this equation as it is not separable.

for an equation in the from $\displaystyle y'+f(x)y = g(x)$

your integrating factor is found as follows

$\displaystyle I = e^{\int f(x) dx} = e^{\int x dx} =e^{ \frac{x^2}{2}}$

multiplying this through the entire equation gives

$\displaystyle e^{ \frac{x^2}{2}}y' + xe^{ \frac{x^2}{2}}y = x^3 e^{ \frac{x^2}{2}}$

then using the product rule, LHS changes to be

$\displaystyle (e^{ \frac{x^2}{2}}y)' = x^3 e^{ \frac{x^2}{2}}$

now integrate both sides

$\displaystyle e^{ \frac{x^2}{2}}y = \int x^3 e^{ \frac{x^2}{2}}dx$

gives y to be

$\displaystyle y = \frac{\int x^3 e^{ \frac{x^2}{2}}dx}{e^{ \frac{x^2}{2}}}$

you will have to finish the problem by using integration by parts on the numerator of the RHS.