zxx+2zxy+zyy=0∂2z∂x2+2∂2z∂x∂y+∂2z∂y2=0(∂∂x+∂∂y)2z=0 ⟹ ∂z∂x+∂z∂y=0 ⟹ dydx=−1 ⟹ y=−x+C, y+x=C ⟹ z=F(y+x)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)zxx+2zxy+zyy=0∂x2∂2z+2∂x∂y∂2z+∂y2∂2z=0(∂x∂+∂y∂)2z=0⟹∂x∂z+∂y∂z=0⟹dxdy=−1⟹y=−x+C,y+x=C⟹z=F(y+x)
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