Help to solve differentail equation

[Mekhala]

eliminate the arbitrary constants a & b from z^2 =ax^3 +by^3+ab

 

[HallsofIvy]

So this is NOT "solving" a differential equation, it is creating a differential equation.

The derivative of a constant is 0 so start differentiating!

Differentiate with respect to x
\(\displaystyle 2z\frac{\partial z}{\partial x}= 3ax^2\)

Differentiate again with respect to y
\(\displaystyle 2z\frac{\partial^2z}{\partial x\partial y}+ 2\frac{\partial z}{\partial x}\frac{\partial z}{\partial y}= 0\)