Solve the system of partial differential equations (∂ z)/(∂…

Solve the system of partial differential equations \[ \frac{\partial z}{\partial x} = \frac{z}{x}, \qquad \frac{\partial z}{\partial y} = \frac{2z}{y}. \]

Step 1 · Integrate the first equation

\[ \frac{dz}{z}=\frac{dx}{x} \;\Longrightarrow\; z = x\,f(y), \tag{1} \] where $f(y)$ is an arbitrary function arising as the constant of integration.

Step 2 · Determine $f(y)$ from the second equation

Substituting $z=xf(y)$ into $z_y=2z/y$: $xf'(y)=2xf(y)/y \Rightarrow f'/f=2/y$. Integrating: $f(y)=Cy^2$.

Step 3 · Compatibility check

The system $z_x=P$, $z_y=Q$ is compatible when $P_y+Pz\cdot0=\partial(z/x)/\partial y = \partial(2z/y)/\partial x$, i.e. \[ \frac{1}{x}z_y = \frac{2}{y}z_x \;\Longrightarrow\; \frac{2z}{xy}=\frac{2z}{xy}, \] which holds identically, confirming the system is compatible for every solution. $\checkmark$

Verification

With $z=Cxy^2$: $z_x=Cy^2=z/x$ $\checkmark$, and $z_y=2Cxy=2z/y$. $\checkmark$