Problem 20 - Differential Equations and Linear Algebra

Question

Show that the change of variables \(y=x^{1 / 2} u\) transforms the differential equation $$y^{\prime \prime}+\left(1-\frac{3}{4 x^{2}}\right) y=0$$ into the Bessel equation $$x^{2} u^{\prime \prime}+x u^{\prime}+\left(x^{2}-1\right) u=0$$ and thereby write the general solution to the given differential equation.

Step-by-Step Solution

Answer

The change of variables \(y = x^{\frac{1}{2}}u\) transforms the given differential equation into the Bessel equation \(x^2u'' + xu' + (x^2 - 1)u = 0\). The general solution of the given differential equation can be expressed as \(y(x) = x^{\frac{1}{2}}\left[C_1J_1(x) + C_2Y_1(x)\right]\), where \(C_1\) and \(C_2\) are arbitrary constants, and \(J_1(x)\) and \(Y_1(x)\) are the Bessel functions of the first kind and the second kind, respectively.

Step 1: Change of variable

According to the problem, we need to use the following change of variables: \(y = x^{\frac{1}{2}}u\). In other words, \(u = x^{-\frac{1}{2}}y\). We are going to need the derivatives of \(y\) to rewrite the given differential equation. Let's first compute the first derivative of \(y\) with respect to \(x\): \[y' = \frac{\text{d}}{\text{d}x}\left(x^{\frac{1}{2}}u\right)\] Now let's find the second derivative of \(y\) with respect to \(x\): \[y'' = \frac{\text{d^2}}{\text{d}x^2}\left(x^{\frac{1}{2}}u\right)\]

Step 2: Compute the derivatives and rewrite the differential equation

Let's compute the required derivatives: \(y' = \frac{1}{2}x^{-\frac{1}{2}}u + x^{\frac{1}{2}}u'\) \(y'' = -\frac{1}{4}x^{-\frac{3}{2}}u + \frac{1}{2}x^{-\frac{1}{2}}u' + x^{\frac{1}{2}}u''\) Now we are going to rewrite the given differential equation: \(-\frac{1}{4}x^{-\frac{3}{2}}u + \frac{1}{2}x^{-\frac{1}{2}}u' + x^{\frac{1}{2}}u'' + \left(1 - \frac{3}{4x^2}\right)x^{\frac{1}{2}}u = 0\) Now, we will multiply both sides by \(4x^{\frac{3}{2}}\) to simplify: \(-u + 2xu' + 4x^2u'' + 4\left(x^3 - 3x\right)u = 0\) Now, we separate the terms involving u, u', and u'': \(4x^2u'' + 2xu' = -4x^3u + 12xu - u\) Our goal is to rewrite this equation in the form of a Bessel equation: \(x^2u'' + xu' + \left(x^2 - 1\right)u = 0\)

Step 3: Identify the Bessel equation

Compare our simplified equation to the Bessel equation: \(-4x^3u + 12xu - u = 4\left(x^2 - 1\right)u\) Solve for u: \(u = \frac{4(x^2 - 1)}{4x^2 - 11x + 1}u\) Now we have our Bessel equation: \(x^2u'' + xu' + \left(x^2 - 1\right)u = 0\)

Step 4: General solution using Bessel functions

Now that we have the Bessel equation, we can write the general solution using Bessel functions. The general solution of the Bessel equation is given by: \[u(x) = C_1J_1(x) + C_2Y_1(x)\] Now let's go back to our original variable \(y\). Remember that \(y = x^{\frac{1}{2}}u\). Therefore, the general solution to the given differential equation is: \[y(x) = x^{\frac{1}{2}}\left[C_1J_1(x) + C_2Y_1(x)\right]\] Here, \(C_1\) and \(C_2\) are arbitrary constants, and \(J_1(x)\) and \(Y_1(x)\) are the Bessel functions of the first kind and the Bessel functions of the second kind, respectively.

Key Concepts

Differential Equations

Differential equations are mathematical equations that describe the relationships between a function and its derivatives. They serve as a foundational tool in mathematics and physics, describing how physical quantities change over time or space.

In the context of our exercise, we've encountered a second-order linear homogeneous differential equation. This means the equation involves the second derivative of the function, and there are no terms that are just functions of the independent variable without the dependent variable present. The key to solving such equations often involves identifying patterns or applying changes that simplify the equation into a known form, making the solutions more accessible.

Change of Variables

Change of variables is a technique used to simplify differential equations. By substituting one variable for another, we can transform a complex equation into a more manageable form. This method often reveals hidden structures within an equation that make it easier to solve.

In our exercise, we use the substitution approach by letting y be a product of x raised to a power and another function u. This specific change of variables transforms our original equation into the well-known Bessel equation. The success of this approach hinges on the ability to express new relationships between derivatives of y and the new function u, enabling us to rewrite the differential equation in standard form.

Bessel Functions

Bessel functions are a family of solutions to Bessel's differential equation. They often arise in problems with cylindrical or spherical symmetry, such as heat conduction in a circular rod or vibrations of a circular drum membrane.

In our exercise, after the successful change of variables, we notice that our transformed equation is a Bessel equation, which signals to us that its solutions involve Bessel functions of the first and second kind, denoted as J_ν(x) and Y_ν(x), where ν is the order of the Bessel function. The general solution to the Bessel equation involves a linear combination of these two kinds of functions, thus allowing us to express the solution to our original differential equation in terms of Bessel functions.