Question

The solution of the pde dz/dx +dz/dy = z^2 is z = -[y+f(x-y)] true or false why?

Answer 1

Answer on Question #66107 – Math – Differential Equations

Question

The solution of the pde dz/dx +dz/dy = z^2 is z = -[y + f(x-y)]. True or false, why?

Solution

If $z = - (y + f(x - y) )$, then

$$\frac{\partial z}{\partial x} = - f' (x - y), \quad \frac{\partial z}{\partial y} = - 1 + f' (x - y).$$

$\frac{\partial z}{\partial x} + \frac{\partial z}{\partial y} = -1 \neq z^2$, hence $z = - (y + f(x - y))$ is not the solution of the partial differential equation $\frac{\partial z}{\partial x} + \frac{\partial z}{\partial y} = z^2$.

Answer: False.

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