Definition used:

The first order linear equation is of the form dydt+p(t)y=q(t) where p and q are function of the independent variable t.

Formula used:

The integrating factor μ(t) of the first order differential equation dydt+p(t)y=q(t) is given as μ(t)=e∫p(t) dt and the solution of the differential equation is given by y=1μ(t)[∫μ(s)q(s)ds+c].

Calculation:

The given differential equation is dydx=x3−2yx.

Rewrite the given differential equation of the form dydt+p(t)y=q(t) as,

dydx=x3−2yxdydx=x3x−2yxdydx+2yx=x2dydx+(2x)y=x2

Note that, p(t)=2x and q(t)=x2.

Compare with the formula and obtain the integrating factor of the differential equation.

μ(x)=e∫2x dx=e2∫1x dx=e2(logx)         (∵∫1x dx=logx)=e(logx)2=x2

Now, obtain the solution as follows.

y=1μ(x)[x+c]=1x2[∫x2(x2)dx+c]=1x2[∫x4dx+c]=1x2[x55+c]=x35+cx2

Therefore, the solution of the given differential equation is y=x35+cx2_.