Question

How to arrange a differential equation as a linear form?

Answer 1

The first thing you need to do is get every term involving y on one side of the equation. So subtracting 8xy from both sides gives you

dydx−8xy=3x2−2x+2.

We now note that the equation is now in the form of a linear equation dydx+P(x)y=Q(x). To proceed from here, you need to compute the integrating factor

μ(x)=exp(∫P(x)dx)=…(I leave this part to you)

where exp(x)=ex. Then if you multiply both sides of the linear ODE by μ(x), you get

ddx[μ(x)y]=μ(x)(3x2−2x+2)⟹y=1μ(x)∫μ(x)(3x2−2x+2)dx.

Can you fill in the work I left out? I hope this helps!