The differential equation dydx2135x18y30xy has an implicit g

The differential equation dy/dx=21+35x+18y+30xy has an implicit general solution of the form F(x,y)=K. In fact, because the differential equation is separable, we can define the solution curve implicitly by a function in the form F(x,y)=G(x)+H(y)=K. Find such a solution and then give the related functions requested.

Solution

Solve this differential equation by separating the variables then integrating:
dy / dx = 21+35x+18y+30xy
dy / dx = 7(3 + 5x) + 6y(3 + 5x)
dy / dx = (3 + 5x)(7 + 6y)
dy / (7 + 6y) = (3 + 5x) dx
? 6 / (7 + 6y) dy / 6 = ? (3 + 5x) dx
ln|7 + 6y| / 6 = 3x + 5x