Answer to Question #226131 in Differential Equations for Hasnain ali

Question

Zxx+2Zxy+Zyy=0

Answer 1

$z_{xx} + 2z_{xy} + z_{yy} = 0\\ \frac{\partial^2 z}{\partial x^2} + 2\frac{\partial^2 z}{\partial x \partial y} + \frac{\partial^2 z}{\partial y^2} = 0\\ \left(\frac{\partial }{\partial x} + \frac{\partial }{\partial y}\right)^2 z = 0\\ \implies \frac{\partial z}{\partial x} + \frac{\partial z}{\partial y} = 0\\ \implies \frac{\mathrm{d}y}{\mathrm{d}x} = -1\\ \implies y = -x + C, \,\, y + x = C\\ \implies z = F(y + x)$