16.11. Method of Separation of Variables

Try these first

1. 1.

   Define the method of separation of variables in your own words.

   Hint

   Consider how we might write the solution format.

2. 2.

   What is the first step in applying this method?

   Hint

   Focus on separating the variables.

3. 3.

   What is the initial assumption in the method of separation of variables?

   • a. u(x,t) = X(x) + T(t)
   • b. u(x,t) = X(x)T(t)
   • c. u(x,t) = X(x)T'(t)

   Hint

   Think about how variables are combined.

4. 4.

   True or False: The method of separation of variables can only be used for second-order PDEs.

   • True
   • False

   Hint

   Consider the flexibility of the method.

5. 5.

   Consider the PDE $\frac{\partial^2 u}{\partial t^2} = \alpha \frac{\partial^2 u}{\partial x^2}$ with boundary conditions $u(0,t) = 0$ and $u(L,t) = 0$. Use the method of separation of variables to determine the general solution.

   Hint

   Follow through with the separation and ensure to use boundary conditions.

6. 6.

   Using the wave equation, solve for $u(x,t)$ given $u(x,0)=f(x)$ and $\frac{\partial u}{\partial t}(x,0)=g(x)$. Assume boundary conditions at both ends are zero.

   Hint

   Consider Fourier series for $f(x)$ and $g(x)$.