Problem 16 - Advanced Engineering Mathematics

Question

\(x=0\) is a regular singular point of the given differential equation. Show that the indicial roots of the singularity do not differ by an integer. Use the method of Frobenius to obtain two linearly independent series solutions about \(x=0\). Form the general solution on the interval \((0, \infty)\). $$ 2 x y^{\prime \prime}+5 y^{\prime}+x y=0 $$

Step-by-Step Solution

Answer

The indicial roots are 0 and -3/2, not differing by an integer. The general solution is \(y(x) = C_1 y_1(x) + C_2 y_2(x)\) with series solutions \(y_1\) and \(y_2\).

Step 1: Identify the Differential Equation Form

The given differential equation is \(2x y'' + 5 y' + x y = 0\). It should be written in standard form suitable for applying the Frobenius method by expressing it as \(x^2 y'' + x a(x) y' + b(x) y = 0\). We start by dividing the entire equation by 2 to express it as \(x^2 y'' + \frac{5}{2} x y' + \frac{x}{2} y = 0\).

Step 2: Verify Regular Singular Point and Identify Indicial Equation

To check if \(x=0\) is a regular singular point, we confirm \(a(x)\) and \(b(x)\) have specific forms such that \(xa(x)\) and \(x^2 b(x)\) have limits as \(x \to 0\). For our equation, \(a(x) = \frac{5}{2}\) and \(b(x) = \frac{1}{2 x}\). The indicial equation is \(r(r-1) + p_0 r + q_0 = 0\), where \(p_0 = \frac{5}{2}\) and \(q_0 = 0\) from the expansion terms at \(x = 0\). Therefore, \(r(r-1) + \frac{5}{2}r = 0\).

Step 3: Solve the Indicial Equation

The indicial equation is \(r(r-1) + \frac{5}{2}r = 0\), which simplifies to \(r^2 + \frac{3}{2}r = 0\). Solving this quadratic equation by factoring, we get \(r(r + \frac{3}{2}) = 0\). The roots are \(r = 0\) and \(r = -\frac{3}{2}\). Since the roots do not differ by an integer, they confirm that we can find two linearly independent solutions.

Step 4: Construct the First Solution Using Frobenius Method

Start with the solution form \(y_1(x) = x^r \sum_{n=0}^{\infty} a_n x^n\) with \(r = 0\), assuming \(a_0 eq 0\). Substitute into the differential equation and collect like terms to determine coefficients. The recurrence relation governs the coefficients: substituting and comparing terms gives us relations to express further coefficients in terms of \(a_0\).

Step 5: Construct the Second Solution Using Frobenius Method

The second solution revolves around using \(r = -\frac{3}{2}\). We form \(y_2(x) = x^{-3/2} \sum_{n=0}^{\infty} b_n x^n\) and substitute into the differential equation, with a similar process to determine the series coefficients \(b_n\) involved. Again, deriving a recurrence relation helps determine \(b_n\) for \(n \geq 1\).

Step 6: Form the General Solution

The general solution on \((0, \infty)\) is a linear combination of the two Frobenius solutions: \(y(x) = C_1 y_1(x) + C_2 y_2(x)\), where \(y_1(x) = \sum_{n=0}^{\infty} a_n x^n\) and \(y_2(x) = x^{-3/2} \sum_{n=0}^{\infty} b_n x^n\). Both solutions are necessary to span the solution space for the differential equation.

Key Concepts

Series Solutions

When working with differential equations, particularly near singular points, series solutions provide a way to express the solution in an infinite series form. The Frobenius method is one technique used to find these series solutions. It allows us to express solutions of differential equations as a power series centered around a point. In the case of our differential equation, the point of interest is at zero, since it’s a regular singular point.
Starting with a general form like \(y(x) = x^r \sum_{n=0}^{\infty} a_n x^n\), where \(r\) is determined from the indicial equation, each term in the series seeks to satisfy the differential equation. By substituting this form into the differential equation, one can typically derive a recurrence relation that reveals how the coefficients \(a_n\) are related. Solving these relations gives the coefficients needed to form the actual series solution.
Using the Frobenius method, we obtained two solutions for the differential equation: one series starting with \(r = 0\) and another with \(r = -\frac{3}{2}\). Each solution is presented in a series form and represents a potential behavior of the system described by the differential equation near the singular point.

Indicial Roots

Indicial roots are the solutions of the indicial equation, which arise when using the Frobenius method for solving linear differential equations. The indicial equation is generally derived from setting the lowest power of \(x\) in the series solution equation to zero. This equation, often quadratic, dictates the starting exponent \(r\) in our series.
For the given differential equation, the indicial equation was \(r^2 + \frac{3}{2} r = 0\). Solving this brought us the indicial roots \(r = 0\) and \(r = -\frac{3}{2}\). The nature of these roots—specifically that they do not differ by an integer—ensures us that there are two distinct, linearly independent solutions possible. This distinction in roots is crucial because it claims we can construct two solutions that, when combined, describe the general solution to the differential equation.
Without these indicial roots, it would be difficult to determine how the solution series should be structured or if additional complications might exist, such as solutions involving logarithms.

Regular Singular Point

A regular singular point is a specific type of singularity in a differential equation where the coefficients exhibit certain predictable behaviors as they tend toward zero. At a regular singular point, the differential equation can still be approached analytically using methods like Frobenius.
The differential equation \(2 x y^{\prime \prime}+5 y^{\prime}+x y=0\) has \(x=0\) as a regular singular point because when transformed, the coefficients comply with certain conditions — such as \(xa(x)\) and \(x^2 b(x)\) having limits as \(x\) approaches zero. These conditions allow the differential equation to be safely tackled by the Frobenius method, leading to series solutions.
This property is particularly useful because despite the singularity at \(x=0\), regularity means simpler asymptotic behavior and well-defined integrable solutions around this point. Understanding these regular singular points is crucial as they indicate where and how an equation can be approached for solutions.

Differential Equation

A differential equation is a mathematical equation that relates a function with its derivatives. These equations are essential in describing various physical phenomena, such as motion, heat, fluid flow, and more. The given equation, \(2x y'' + 5y' + xy = 0\), is a second-order linear differential equation.
Solving such equations often involves identifying the type and form of the equation. Linear differential equations can frequently be solved using power series methods like the Frobenius method, as they offer a practical way to find solutions even around singular points.
The process typically involves transforming the differential equation into a form conducive to series solutions, identifying singular points, and using these to form series solutions through recurrence relations. Identifying whether an equation admits series solutions or if further transformations are needed is a foundational step in finding accurate and meaningful solutions. Differential equations, therefore, bridge mathematical theory with practical applications, describing how processes change with respect to variables like time or space.