The given partial differential equation (PDE) is:

$\frac{\partial u}{\partial x} \cdot \frac{\partial u}{\partial y} = u(x, y)$

with the boundary conditions:

1. $u(x, 0) = 0$
2. $u(0, y) = 1$

We are tasked with solving this using the separation of variables method.

Solution Steps:

1. Assume $u(x, y)$ can be written as a product of two functions:

Let: $u(x, y) = X(x)Y(y)$

2. Substitute $u(x, y) = X(x)Y(y)$ into the PDE:

$\frac{\partial u}{\partial x} = X'(x)Y(y), \quad \frac{\partial u}{\partial y} = X(x)Y'(y)$

Substituting these into the given PDE: $X'(x)Y(y) \cdot X(x)Y'(y) = X(x)Y(y)$

Simplify: $X'(x)X(x) \cdot Y'(y)Y(y) = X(x)Y(y)$

Divide through by $X(x)Y(y)$ (assuming $X(x)Y(y) \neq 0$): $\frac{X'(x)}{X(x)} \cdot \frac{Y'(y)}{Y(y)} = 1$

3. Separate the variables:

Since the left-hand side depends only on $x$ and $y$ independently, set each side equal to a separation constant, $\lambda$: $\frac{X'(x)}{X(x)} = \lambda, \quad \frac{Y'(y)}{Y(y)} = \frac{1}{\lambda}$

4. Solve the ordinary differential equations (ODEs):

• For $X(x)$: $\frac{X'(x)}{X(x)} = \lambda \implies X'(x) = \lambda X(x)$ $X(x) = C_1 e^{\lambda x}$

• For $Y(y)$: $\frac{Y'(y)}{Y(y)} = \frac{1}{\lambda} \implies Y'(y) = \frac{1}{\lambda} Y(y)$ $Y(y) = C_2 e^{\frac{y}{\lambda}}$

Thus, the general solution is: $u(x, y) = X(x)Y(y) = C_1C_2 e^{\lambda x} e^{\frac{y}{\lambda}}$

Let $C = C_1C_2$, so: $u(x, y) = C e^{\lambda x} e^{\frac{y}{\lambda}}$

5. Apply the boundary conditions:

• From $u(x, 0) = 0$: $u(x, 0) = C e^{\lambda x} e^{\frac{0}{\lambda}} = C e^{\lambda x} = 0$

This implies $C = 0$, which is trivial and not useful. We need a different approach to handle the boundary conditions. Instead, this suggests a refinement of separation constants or additional conditions on $\lambda$.

Would you like me to refine this further, or proceed with an alternative solution approach?

Related Questions:

1. What is the method of separation of variables, and why is it effective for solving PDEs?
2. How do boundary conditions influence the general solution of a PDE?
3. Can the separation of variables method be extended for non-linear PDEs like this one?
4. What alternative methods exist for solving PDEs with given boundary conditions?
5. How do we interpret the physical meaning of $\lambda$ in such problems?

Tip: Always check the consistency of your boundary conditions before solving to avoid trivial solutions.