Question

Find a partial differential equation by eliminating a and b from the equations

z^2 = ax^3 + by^3 + ab.

Answer 1

Let us find a partial differential equation by eliminating $a$ and $b$ from the equations $z^2 = ax^3 + by^3 + ab.$

Let us use the implicit differentiation.

It follows that

$2zz_x = 3ax^2$ and $2zz_y = 3by^2.$

Therefore, $a=\frac{2zz_x}{ 3x^2}$ and $b=\frac{2zz_y}{ 3y^2}.$

We conclude that the partial differential equation is of the form:

$z^2 = \frac{2zz_x}{ 3x^2}x^3 + \frac{2zz_y}{ 3y^2}y^3 + \frac{2zz_x}{ 3x^2}\frac{2zz_y}{ 3y^2}.$