# 2.2: The Method of Separation of Variables

- Page ID
- 13376

Learning Objectives

- To be introduced to the Separation of Variables as method to solved wave equations

Solving the wave equation involves identifying the functions \(u(x,t)\) that solve the partial differential equation that represent the amplitude of the wave at any position \(x\) at any time \(t\)

\[ \dfrac{\partial^2 u(x,t)}{\partial x^2} = \dfrac{1}{v^2} \dfrac{\partial^2 u(x,t)}{\partial t^2} \label{2.1.1}\]

This wave equation is a type of second-order partial differential equation (PDE) involving two variables - \(x\) and \(t\). PDEs differ from ordinary differential equations (ODEs) that involve functions of only one variable. However, this difference makes PDEs appreciably more difficult to solve. In fact, the vast majority of PDE cannot be solved analytically and those classes of special PDEs that can be solved analytically invariably involve converting the PDE into one or more ODEs and then solving independently. One of these approaches is the the method of *separation of variables*.

Method of Separation of Variables

The general application of the *Method of Separation of Variables* for a wave equation involves three steps:

- We find all solutions of the wave equation with the general form \[u(x,t)= X(x)T(t)\]for some function \(X(x)\) that depends on \(x\) but not \(t\) and some function \(T(t)\) that depends only on \(t\), but not \(x\). It is of course too much to expect that all solutions of Equation \(\ref{2.1.1}\) are of this form, however, if we find a set of solutions \(\{X_i(x)T_i(t)\}\) since the wave equation is a
*linear equation*, \[u(x,t)=\sum_i c_ iX_i(x)T_i(t) \label{gen1}\]is also a solution for*any choice*of the constants \(c_i\). - Impose constraints on the solutions based on the knowledge of the system. These are called the
**boundary conditions**, which specify the values of \(u(x,t)\) at the extremes ("boundaries"). This is a similar constraint to the solution as in initial value problems which the conditions \(x(t_i)\) are specified at a specific time \(t_i\). The goal is then to select the constants \(c_i\) in Equation \ref{gen1} so that the boundary conditions are also satisfied.

Method of separation of variables is one of the most widely used techniques to solve partial differential equations and is based on the *assumption *that the solution of the equation is separable, that is, the final solution can be represented as a product of several functions, each of which is only dependent upon a single independent variable. If this assumption is incorrect, then clear violations of mathematical principles will be obvious from the analysis.

## A Vibrating Spring Held Fixed Between Two Points

As discussed in Section 2.1, the solutions to the string example \(u(x,t)\) for all \(x\) and \(t\) would be assumed to be a product of two functions: \(X(x)\) and \(T(t)\), where \(X(x)\) is a function of only \(x\), not \(t\) and \(T(t)\) is a function of \(t\), but not \(x\).

\[u(x,t)= X(x)T(t) \label{2.2.1}\]

Substitute Equation \(\ref{2.2.1}\) into the one-dimensional wave equation (Equation \(\ref{2.1.1}\)) gives

\[ \dfrac{\partial^2 X(x)T(t)}{\partial x^2} = \dfrac{1}{v^2} \dfrac{\partial^2 X(x)T(t)}{\partial t^2} \label{2.2.2}\]

Since \( X \) is not a function of \(t\) and \(T\) is not a function of \(x\), Equation \(\ref{2.2.2}\) can be simplified

\[ T(t) \dfrac{\partial^2 X(x)}{\partial x^2} = \dfrac{1}{v^2} X(x) \dfrac{\partial^2T(t)}{\partial t^2} \label{2.2.3}\]

Collecting the expressions that depend on \(x\) on the left side of Equation \(\ref{2.2.3}\) and of \(t\) on the right side results in

\[ \dfrac{1}{X(x)} \dfrac{\partial^2 X(x)}{\partial x^2} = \dfrac{1}{v^2} \dfrac{1}{T(t)} \dfrac{\partial^2T(t)}{\partial t^2} \label{2.2.3a}\]

Equation \(\ref{2.2.3a}\) is an interesting equation since the each side can be set to a fixed constant \(K\) as that is the only solution that works for all values of \(t\) and \(x\). Therefore, the equation can be separated into two *ordinary differential equations:*

\[ \dfrac{d^2T(t)}{dt^2} - Kv^2 T(t) = 0 \label{2.2.4a}\]

\[\dfrac{d^2X(x)}{dx^2} - K X(x) = 0 \label{2.2.4b}\]