Question

Form the partial differential equation by eliminating the arbitrary constants a and b from z = xy + y((x+a) ^1/2) + b

Answer 1

$z=xy+y\sqrt{x+a}+b\\\left\{ \begin{array}{c} z'_x=y+\frac{y}{2\sqrt{x+a}}\\ z'_y=x+\sqrt{x+a}\\\end{array} \right. \Rightarrow \left\{ \begin{array}{c} \sqrt{x+a}=z'_y-x\\ z'_x=y+\frac{y}{2\left( z'_y-x \right)}\\\end{array} \right. \Rightarrow \\\Rightarrow z'_x=y+\frac{y}{2\left( z'_y-x \right)}$