Problem 28 - Advanced Engineering Mathematics

Question

\(x=0\) is a regular singular point of the given differential equation. Show that the indicial roots of the singularity differ by an integer. Use the method of Frobenius to obtain at least one series solution about \(x=0 .\) Use \((21)\) where necessary and a CAS, if instructed, to find a second solution. Form the general solution on the interval \((0, \infty)\). $$ y^{\prime \prime}+\frac{3}{x} y^{\prime}-2 y=0 $$

Step-by-Step Solution

Answer

The roots are \( r = -1 \pm \sqrt{3} \), differing by a nontrivial integer.

Step 1: Identify the Equation Type

The given differential equation is of the form \( y'' + \frac{P(x)}{x} y' + \frac{Q(x)}{x^2} y = 0 \) with \( P(x) = 3 \) and \( Q(x) = -2x^2 \). This classifies the equation as a regular singular point at \( x = 0 \).

Step 2: Set Up the Indicial Equation

The indicial equation is derived from the lowest power of the series solution at the singular point. For the differential equation in the form \( x^2 y'' + x P(x) y' + Q(x) y = 0 \), the indicial equation is \( r(r-1) + rP_0 + Q_0 = 0 \), where \( P_0 = 3 \) and \( Q_0 = 2 \).

Step 3: Solve the Indicial Equation

Substitute \( P_0 = 3 \) and \( Q_0 = -2 \) into the indicial equation: \( r(r-1) + 3r - 2 = 0 \). Simplify to find \( r^2 + 2r - 2 = 0 \).

Step 4: Find the Indicial Roots

Solve the quadratic equation \( r^2 + 2r - 2 = 0 \) using the quadratic formula: \( r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) where \( a = 1, b = 2, c = -2 \). This results in \( r = -1 \pm \sqrt{3} \). The roots differ by an integer.

Step 5: Form the Frobenius Solution

Since the roots differ by an integer, use the larger root \( r = -1 + \sqrt{3} \) to construct the series solution \( y_1(x) = \sum_{n=0}^{\infty} a_n x^{n+r} \). Substitute into the differential equation and solve for the coefficients \( a_n \).

Step 6: Use a CAS for Second Solution

The second solution can be found using a Computer Algebra System (CAS) because the roots do not have a simple integer difference. The CAS uses the first solution \( y_1(x) \) and generates \( y_2(x) \) using variation of parameters.

Step 7: Form the General Solution

Combine both series solutions to form the general solution: \( y(x) = c_1 y_1(x) + c_2 y_2(x) \), which is valid on the interval \((0, \infty)\).

Key Concepts

Indicial Equation

When dealing with differential equations at a singular point, especially using the Frobenius method, we encounter the indicial equation. This is a key step that helps us determine the nature of solutions near a singularity. In the context of a differential equation, the indicial equation is derived from the substitution of a power series into the equation's lowest power term.
For example, consider a differential equation of the form:

• \[ y'' + \frac{P(x)}{x} y' + \frac{Q(x)}{x^2} y = 0 \]

At a singular point, the indicial equation allows us to handle terms that otherwise become undefined at the singularity. Solving it gives us the indicial roots, which tell us the exponents in the leading terms of the series solutions.

Series Solution

The series solution method involves assuming a solution in the form of a power series, expanding the function around a point. This method is especially useful for solving linear differential equations with variable coefficients. In these cases, solutions are often not expressed in terms of elementary functions. Instead, they can be written more naturally with series solutions.
In the Frobenius method, after solving the indicial equation, we use the leading root \(r\) as the starting point for our series. The solution generally looks like:

• \[ y(x) = \sum_{n=0}^{\infty} a_n x^{n+r} \]

This approach helps find recurrent coefficients \(a_n\) by substituting back into the differential equation and aligning terms with similar powers of \(x\). This process is iterated to generate a solution step-by-step.

Differential Equations

Differential equations are mathematical equations involving derivatives, which represent how a quantity changes. They are fundamental in expressing laws of various fields such as physics, engineering, and economics. There are different types of differential equations, such as ordinary, partial, linear, or nonlinear.
A second-order linear differential equation, like the one in this exercise, takes the form:

• \[ y'' + P(x) y' + Q(x) y = 0 \]

Specific characteristics, like singular points, influence how these equations are handled. To solve them, especially around singular points, specialized methods like Frobenius are used, allowing us to explore complex behavior and accommodations for non-constant coefficient scenarios.

Regular Singular Point

A regular singular point is a specific type of singularity in differential equations. At these points, solutions might become infinite, or certain expressions might become undefined. However, they provide more manageable behavior than irregular singular points.
An equation of the standard form:

• \[ x^2 y'' + x P(x) y' + Q(x) y = 0 \]

will have a regular singularity if the functions \(P(x)\) and \(Q(x)\) are analytic at \(x = 0\). The presence of a regular singular point justifies the use of the Frobenius method, as it ensures that a series solution exists. This property helps ensure that even though simple substitution might fail, we can still find solutions effectively through the indicial equation and subsequent series expansion.