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

Short answer

If so, what is the resulting pair of ordinary differential equations? **Answer:** Yes, the method of separation of variables can be applied to the given partial differential equation. The resulting pair of ordinary differential equations are: 1. \(X''(x) + k^2X(x) = 0\) 2. \(T'(t) - k^2 x T(t) = 0\)

Step-by-step solution

Assume a solution form

Start by assuming a solution of the form \(u(x,t) = X(x)T(t)\).

Differentiate the assumed solution

Find the second partial derivative of \(u(x,t)\) with respect to \(x\) and the first partial derivative of \(u(x,t)\) with respect to \(t\), using the product rule for differentiation. We have: $$u_{xx}(x,t) = X''(x)T(t)$$ and $$u_{t}(x,t) = X(x)T'(t)$$

Plug the derivatives into the PDE

Substitute the expressions for \(u_{xx}(x,t)\) and \(u_{t}(x,t)\) into the given PDE: $$x(X''(x)T(t)) + X(x)T'(t) = 0$$

Separate the variables

Divide through by \(xXT\): $$\frac{X''(x)}{X(x)} + \frac{T'(t)}{x \cdot T(t)} = 0$$

Form two separate ODEs

Since the equation now consists of a pure function of \(x\) on the left and another pure function of \(t\) on the right, we can form two separate ODEs. Set the \(x\)-dependent part of the equation equal to a constant \(-k^2\) and the \(t\)-dependent part of the equation to the same constant \(-k^2\): $$\frac{X''(x)}{X(x)} = -k^2$$ and $$\frac{T'(t)}{x \cdot T(t)} = k^2$$ The pair of ordinary differential equations obtained by applying the method of separation of variables to the given PDE are: $$X''(x) + k^2X(x) = 0$$ and $$T'(t) - k^2 x T(t) = 0$$

Key concepts

Partial Differential Equation

A Partial Differential Equation (PDE) involves multiple independent variables and their partial derivatives. It's a fundamental concept in mathematics and physics, as it describes phenomena that vary across multiple dimensions. For instance, the heat equation, wave equation, and Laplace's equation are common PDEs that model physical processes.

More formally, a PDE is an equation involving functions and their partial derivatives. For example, the equation from the textbook exercise \(x u_{xx} + u_t = 0\) involves a function \(u(x, t)\), which depends on two independent variables: \(x\) for space and \(t\) for time. The subscripts \(u_{xx}\) and \(u_t\) denote the second partial derivative with respect to \(x\) and the first partial derivative with respect to \(t\), respectively. Solving PDEs often requires sophisticated methods, as they represent complex systems with numerous dynamic components.

Ordinary Differential Equations

Ordinary Differential Equations (ODEs) are equations that contain one or more functions of one independent variable and its derivatives. They are called 'ordinary' to distinguish them from partial differential equations, which involve partial derivatives of multiple independent variables. ODEs are foundational in the mathematical modelling of natural phenomena, from celestial mechanics to biological systems.

As an example, the motion of a pendulum over time can be described by a second-order ODE that relates the angle of displacement with respect to time. In the context of the exercise, once the method of separation of variables is applied, the PDE is converted into two simpler ODEs, which are easier to solve and analyze.

Method of Separation of Variables

The method of separation of variables is a powerful technique for solving certain PDEs. The fundamental idea is to transform a complex PDE into simpler, separate ODEs that depend on only one variable each. This is achieved by assuming that the solution can be represented as the product of functions that each depends on a single coordinate.

For the given exercise, a product solution of the form \(u(x,t) = X(x)T(t)\) is assumed. The separate functions \(X\) and \(T\) represent spatial and temporal components respectively. After appropriate manipulation and application of this method, two independent ODEs are derived. The success of this method relies on the ability to isolate variables to either side of the equation, which often involves clever algebraic techniques and insightful assumptions.

Product Rule for Differentiation

The product rule is a basic rule in differential calculus used to find the derivative of the product of two functions. It states that the derivative of a product is equal to the derivative of the first function times the second function, plus the first function times the derivative of the second function. Mathematically, if \(f\) and \(g\) are functions, then their product's derivative \(f\times g\) is given by \(f' g + f g'\).

In the context of our exercise, the product rule is essential for differentiating \(u(x,t) = X(x)T(t)\) with respect to \(x\) and \(t\). This differentiation produces the expressions needed to substitute back into the original PDE and facilitates the separation of variables process.