1. **State the problem:** Solve the partial differential equation (PDE) given by $$n \frac{\partial u}{\partial m} = y \frac{\partial u}{\partial y}$$ where $u = u(m,y)$. 2. **Assume a separable solution:** Let $u(m,y) = X(m) Y(y)$. 3. **Substitute into the PDE:** $$n X'(m) Y(y) = y X(m) Y'(y)$$ Divide both sides by $X(m) Y(y)$: $$n \frac{X'(m)}{X(m)} = y \frac{Y'(y)}{Y(y)}$$ 4. **Separate variables:** Since the left side depends only on $m$ and the right side only on $y$, both sides equal a constant $k$: $$n \frac{X'}{X} = k \quad \Rightarrow \quad \frac{X'}{X} = \frac{k}{n}$$ $$y \frac{Y'}{Y} = k \quad \Rightarrow \quad \frac{Y'}{Y} = \frac{k}{y}$$ 5. **Solve the ODEs:** - For $X(m)$: $$\frac{dX}{dm} = \frac{k}{n} X \implies X(m) = C_1 e^{\frac{k}{n} m}$$ - For $Y(y)$: $$\frac{dY}{dy} = \frac{k}{y} Y \implies Y(y) = C_2 y^{k}$$ 6. **Write the general solution:** $$u(m,y) = C e^{\frac{k}{n} m} y^{k}$$ where $C = C_1 C_2$ is an arbitrary constant. 7. **Eigenvalues:** The constant $k$ can take values $0$, $d$, or $-d$ as given. **Final answer:** $$u(m,y) = C e^{\frac{k}{n} m} y^{k} \quad \text{for} \quad k = 0, d, -d$$