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. $$ t u_{x x}+x u_{t}=0 $$

Short answer

Question: Determine if the partial differential equation $tu_{xx} + xu_t = 0$ can be separated into a pair of ordinary differential equations (ODEs) using the method of separation of variables. If possible, find the pair of ODEs. Answer: Yes, the given partial differential equation can be separated into a pair of ordinary differential equations using the method of separation of variables. The pair of ODEs obtained are: $$ X''(x)-kX(x)=0 $$ and $$ \frac{1}{T(t)} T'(t)=-kt^2 $$

Step-by-step solution

Assume a separable solution

We start by assuming that the solution to the given PDE can be written in the form: $$ u(t, x)=T(t)X(x) $$ where T(t) is a function of t only, and X(x) is a function of x only.

Compute partial derivatives

Now we compute the derivatives of u(t, x) with respect to each variable: 1. The second partial derivative of u(t, x) with respect to x: $$ u_{xx}(t, x) = T(t)X''(x) $$ 2. The first partial derivative of u(t, x) with respect to t: $$ u_t(t, x) = T'(t)X(x) $$

Plug derivatives into the given PDE

Next, we substitute the computed derivatives into the given partial differential equation: $$ t(T(t)X''(x)) + x(T'(t)X(x)) = 0 $$

Rewrite the equation

Rewrite the above equation as: $$ T(t)X''(x)+\frac{x}{t}T'(t)X(x) = 0 $$

Separate variables

Separate the variables t and x: $$ \frac{1}{X(x)} X''(x)=-\frac{x}{t}\frac{1}{T(t)} T'(t) $$ At this point, we have succeeded in separating the variables t and x. The left side of the equation depends only on x, while the right side depends only on t. And since both expressions are equal, they must be equal to a constant, say k.

Set up the resulting ODEs

We now set each side equal to the constant k and solve for the ODE in each variable: 1. For x: $$ \frac{1}{X(x)} X''(x)=k $$ which results in the ODE $$ X''(x)-kX(x)=0 $$ 2. For t: $$ -\frac{x}{t}\frac{1}{T(t)} T'(t)=k $$ which can be written as $$ \frac{1}{T(t)} T'(t)=-kt $$ Dividing by t yields the ODE $$ \frac{1}{T(t)} T'(t)=-kt^2 $$ This completes the separation of variables. The pair of ODEs obtained are: $$ X''(x)-kX(x)=0 $$ and $$ \frac{1}{T(t)} T'(t)=-kt^2 $$

Key concepts

Separation of Variables

Separation of Variables is a powerful mathematical technique used to solve certain partial differential equations (PDEs). It works by assuming that the solution to a PDE can be expressed as a product of single-variable functions. In other words, we separate the variables by writing the solution as a product of functions, where each function depends on only one of the variables involved. For example, in our provided exercise, the solution was expressed as \( u(t, x) = T(t)X(x) \). Here, \( T(t) \) is a function that depends solely on time \( t \), and \( X(x) \) is a function that depends solely on space \( x \).

The key idea is that by making this assumption, we can transform the original PDE into simpler ordinary differential equations (ODEs), one for each variable. This is done by differentiating the assumed solution and substituting these derivatives back into the original PDE. What we are aiming for is to make the PDE into separate equations, each involving only one variable. Once separated, these equations can be independently solved using methods appropriate for ODEs.

Ordinary Differential Equations

Ordinary Differential Equations (ODEs) are equations involving functions of a single variable and their derivatives. Once a PDE has been transformed into separate equations using the separation of variables technique, we are left with ODEs which are simpler to handle.

In the exercise, after the separation of variables was applied, we derived two ODEs: one in terms of \( x \) and another in terms of \( t \). The equation \( X''(x) - kX(x) = 0 \) is an ODE in terms of the spatial variable \( x \), and its solution involves functions that depend on \( x \) only. Likewise, the equation \( \frac{1}{T(t)} T'(t) = -kt^2 \) is an ODE in terms of the temporal variable \( t \).

The ODEs each represent a part of the overall solution and can be solved using standard methods such as characteristic equations, integrating factors, or even directly solving them when the equations are simple enough. This process transforms solving a complex PDE into solving simpler, more familiar problems.

Separable Solutions

Separable Solutions refer to the solutions to PDEs that can be broken down into simpler forms through the technique of Separation of Variables. These solutions rely on the PDE being written in a manner where you can assign each of the independent variables an individual function.

In the step-by-step solution to our exercise, the PDE was successfully split into two separate ODEs for the functions \( X(x) \) and \( T(t) \). The form of these ODEs indicates that solutions to the original PDE can be expressed as a product of the solutions to these ODEs, so \( u(t, x) = T(t)X(x) \). This separability often indicates underlying symmetry or conservation properties in the modeled system.

Separable solutions are not only elegant but also provide deep insights into the behavior of dynamic systems. They offer a clear view of how changes in one variable affect the entire solution, allowing complex multidimensional problems to be simplified significantly.