Answer to Question #183129 in Differential Equations for sindhu priya

Question

z= xy+f(x^2+y^2+z^2)

Answer 1

$\displaystyle z = xy + f(x^2 + y^2 + z^2)\\ \frac{\partial z}{\partial x} = y + 2xf(x^2 + y^2 + z^2)\\ \frac{\partial z}{\partial y} = x + 2yf(x^2 + y^2 + z^2)\\ y\frac{\partial z}{\partial x} = y^2 + 2xyf(x^2 + y^2 + z^2)\\ x\frac{\partial z}{\partial y} = x^2 + 2xyf(x^2 + y^2 + z^2)\\ y\frac{\partial z}{\partial x} - x\frac{\partial z}{\partial y} = y^2 - x^2$