Question

Explain the method of separation of variables with an example.

Answer 1

- The **method of separation of variables** is used to solve PDEs by expressing the solution as a product of two functions, each depending on only one variable.

#### **Example: Solve**  
\[
\frac{\partial u}{\partial t} = k \frac{\partial^2 u}{\partial x^2}
\]

1. **Assume a solution of the form:**  
   \[
   u(x,t) = X(x)T(t)
   \]

2. **Substituting in the given equation:**
   \[
   X(x) \frac{dT}{dt} = k T(t) \frac{d^2X}{dx^2}
   \]

3. **Separating the variables:**
   \[
   \frac{1}{T} \frac{dT}{dt} = k \frac{1}{X} \frac{d^2X}{dx^2} = -\lambda
   \]

4. **Solving for \( X(x) \) and \( T(t) \):**  
   \[
   \frac{d^2X}{dx^2} + \lambda X = 0
   \]
   \[
   \frac{dT}{dt} + k\lambda T = 0
   \]

   The solutions are:
   \[
   X(x) = A \cos \sqrt{\lambda} x + B \sin \sqrt{\lambda} x
   \]
   \[
   T(t) = C e^{-k\lambda t}
   \]

5. **Final solution:**  
   \[
   u(x,t) = (A \cos \sqrt{\lambda} x + B \sin \sqrt{\lambda} x) e^{-k\lambda t}
   \]