Question

Determine whether the method of separation of variables can be used to replace the given partial differential equation by a pair of ordinary differential equations. If so, find the equations. $$ \left[p(x) u_{x}\right]_{x}-r(x) u_{t t}=0 $$

Short answer

If so, find the ordinary differential equations resulting from the separation of variables. Answer: Yes, the method of separation of variables can be applied to the given partial differential equation. The resulting ordinary differential equations are: $$ p(x)X''(x) + p'(x)X'(x) = \lambda X(x) $$ and $$ r(x)T''(t) = \lambda T(t) $$

Step-by-step solution

Substitute the Separation of Variables assumption

Using the assumption \(u(x, t) = X(x)T(t)\), compute the required derivatives: - \(u_x = X'(x)T(t)\) - \(u_{xx} = X''(x)T(t)\) - \(u_{tt} = X(x)T''(t)\) Substitute these expressions back into the given PDE: $$ [p(x)X'(x)T(t)]_x - r(x)X(x)T''(t) = 0 $$

Differentiate the Expression

Now, differentiate the term inside the brackets with respect to \(x\): $$ [p(x)X'(x)]_x = p'(x)X'(x) + p(x)X''(x) $$ Substitute this back into our PDE to obtain: $$ [p'(x)X'(x)T(t) + p(x)X''(x)T(t)] - r(x)X(x)T''(t) = 0 $$

Separate the Functions

Now, divide the equation by \(X(x)T(t)\): $$ \frac{p'(x)X'(x)}{X(x)} + \frac{p(x)X''(x)}{X(x)} - \frac{r(x)T''(t)}{T(t)} = 0 $$

Create the Ordinary Differential Equations

Observe that the terms on the left-hand side depend only on \(x\) and the term on the right-hand side depends only on \(t\). For the equation to hold true for all \(x\) and \(t\), both of these terms must be constant. Let's call this constant \(-\lambda\). Since both expressions are negative, we obtain the following ODEs: $$ \frac{p'(x)X'(x)}{X(x)} + \frac{p(x)X''(x)}{X(x)} = \lambda $$ and $$ \frac{r(x)T''(t)}{T(t)} = \lambda $$ Rearranging, we have the resulting ODEs: $$ p(x)X''(x) + p'(x)X'(x) = \lambda X(x) $$ and $$ r(x)T''(t) = \lambda T(t) $$ Thus, by using the separation of variables, we have replaced the given PDE with a pair of ODEs.

Key concepts

Partial Differential Equations

Partial differential equations (PDEs) represent a class of equations used to describe a wide array of phenomena, such as sound, heat, electrostatics, electrodynamics, fluid flow, and many others. In essence, PDEs involve functions of multiple variables and their partial derivatives.

When we refer to a PDE, we're dealing with an equation that contains partial derivatives—derivatives with respect to one variable while holding the others constant. The PDE from our exercise, \[ \left[p(x) u_{x}\right]_{x}-r(x) u_{t t}=0 \], is an example. It tells us that a certain combination of the partial derivatives of a function, in this case, \( u(x, t) \), and the function itself, along with the functions \( p(x) \) and \( r(x) \) which may describe properties of a medium or space, results in zero.

Due to their complexity, solving PDEs is often a central challenge in applied mathematics and physics. Simplifying these equations through techniques such as separation of variables can transform PDEs into more manageable ordinary differential equations (ODEs), which offer a step towards finding a solution.

Ordinary Differential Equations

Ordinary differential equations (ODEs) are equations that relate a function to its derivatives; however, unlike PDEs, the functions in ODEs depend on a single independent variable. They serve as the cornerstone for classical mechanics, quantum mechanics, general relativity, and many other theoretical frameworks.

For a given ODE, the primary goal is often to find a function that satisfies the equation, which is known as the solution to the ODE. The process of solving involves integrating, which essentially is finding the antiderivative or the original function before differentiation occurred.

Take, for instance, the pair of ODEs we derived from the original PDE: \[ p(x)X''(x) + p'(x)X'(x) = \lambda X(x) \] and \[ r(x)T''(t) = \lambda T(t) \]. These equations are now in a form where traditional ODE-solving techniques, including characteristic equations, integrating factors, or in some cases, power series, can be applied to find explicit solutions \( X(x) \) and \( T(t) \), that are functions of a single variable \( x \) and \( t \), respectively.

Mathematical Methods for Differential Equations

Solving differential equations, be they partial or ordinary, requires a toolbox of mathematical methods. Some of the most common strategies include separation of variables, method of characteristics, integral transforms (like Fourier or Laplace transforms), and numerical methods (such as Euler's method or Runge-Kutta methods).

Separation of variables, the method we employed in the textbook exercise, is one effective strategy for solving equations where the solution can be separated into functions that each depend on only one of the variables. By carefully manipulating the equation, we can segregate the components involving different variables onto different sides of the equation, as we did when we obtained \[ \frac{p'(x)X'(x)}{X(x)} + \frac{p(x)X''(x)}{X(x)} = \lambda \] and \[ \frac{r(x)T''(t)}{T(t)} = \lambda \].

After separation, the challenge becomes solving the resultant ODEs for each variable independently. In some cases, these resulting ODEs can be standard forms for which known solutions exist, or they may require further inventive strategies to solve. For a student looking to grasp these topics, practicing these methods on various differential equations is invaluable, as it builds familiarity with the approaches and their applications to different types of problems.