Write the equation in the form y′=f(y/x) then use the substitution y=xu to find an implicit general solution. Then solve the initial value problem.

y′=5y2+4x2xy, y(1)=3

The resulting differential equation in x x and u u can be written as xu′= which is separable.

Separating variables we arrive at du=dx/x

Separating variables and and simplifying the solution can be written in the form u2+1=Cf(x) where C is an arbitrary constant and f(x)= .

Transforming back into the variables x and y and using the initial condition to find C we find the explicit solution of the initial value problem is y = .