When we solve the most basic differential equations, what we do is to seperate x and y parts then do integration. But there are times that we can't seperate the x and y parts perfectly, and in that case we will need to find implicit solution to the differential equations. Here I will provide a simple example on how to do this, and actually for more complicated problems it works the same way.

Example: Find an implict solution to (2y-y^4)dy/dx - x^3 = 0

Solution: We will first multiply both sides by dx, so we can get this:

(2y-y^4)dy - x^3 dx  = 0

then:

(2y-y^4)dy = x^3 dx

Now just integrate and we will get this:

y^2 - y^5 / 5  = x^4 /4 + C

 

(Because the question did not provide conditions for C, so we will just leave the constant C there as part of the answer. If you are given conditions for C, just plug it into the equation and solve for C at the end)