Problem 30 - 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 \((0, \infty)\). $$ x y^{\prime \prime}+y^{\prime}+y=0 $$

Step-by-Step Solution

Answer

The indicial roots are \( r_1 = 0 \) and \( r_2 = 1 \), differing by 1. A series solution is \( y_1(x) = 1 \). General solution is \( y(x)=C_1y_1+C_2y_2 \).

Step 1: Identify the Differential Equation Type

The differential equation given is: \( x y'' + y' + y = 0 \). Identify it as a linear second-order differential equation with a singular point at \( x = 0 \).

Step 2: Find Indicial Equation Using Frobenius Method

Assume a solution of the form \( y = \sum_{n=0}^{\infty} a_n x^{n+r} \) around the singularity \( x=0 \). This method is used to find a series solution at a regular singular point.

Step 3: Define Series Derivatives Equation

Define the series expansions for \( y' \) and \( y'' \): \( y' = \sum_{n=0}^{\infty} a_n (n+r) x^{n+r-1} \) and \( y'' = \sum_{n=0}^{\infty} a_n (n+r)(n+r-1) x^{n+r-2} \).

Step 4: Substitute Series into the Differential Equation

Substitute the series expansions of \( y \), \( y' \), and \( y'' \) into the differential equation \( x y'' + y' + y = 0 \). This results in \( x \sum_{n=0}^{\infty} a_n (n+r)(n+r-1) x^{n+r-2} + \sum_{n=0}^{\infty} a_n (n+r) x^{n+r-1} + \sum_{n=0}^{\infty} a_n x^{n+r} = 0 \). Factor and simplify the terms.

Step 5: Simplify and Equate Powers of x

Simplify the equation by equating coefficients of like powers of \( x \) to zero. This provides the equivalent series solved term-by-term.

Step 6: Form Indicial Equation

The leading coefficient of the lowest power of \( x \) yields the indicial equation. For the above equation, the indicial equation simplifies to \( r^2 - r + a_0 = 0 \) using \( x^{-1} \) term. Solve for \( r \).

Step 7: Solve the Indicial Equation

Solve \( r^2 - r = 0 \), which factors to \( r(r-1) = 0 \), giving roots \( r_1 = 0 \) and \( r_2 = 1 \). The roots differ by an integer.

Step 8: Find the First Series Solution

Substitute \( r = 0 \) into the series to find the first solution. Given \( a_0 eq 0 \), assume \( a_0=1 \) and determine coefficients recursively ensuring higher terms' contributions are zero-filled by recurrence relation. This yields \( y_1 = a_0 = 1 \).

Step 9: Find the Second Series Solution Using CAS

Use a CAS or theoretical methods for \( r = 1 \), substituting into the recurrence relations from Step 8. Solving gives the second independent series solution \( y_2 eq y_1 \).

Step 10: Construct General Solution using Superposition

Since the solutions are independent, the general solution on interval \((0, \infty)\) is \( y(x) = C_1 y_1(x) + C_2 y_2(x) \) where \( C_1 \) and \( C_2 \) are constants.

Key Concepts

Regular Singular Point

In the realm of second-order differential equations, a regular singular point is a specific type of singularity that allows us to find solutions using the Frobenius Method. The equation given, \(x y'' + y' + y = 0\), has a singular point at \(x = 0\). It's termed 'regular' because when you rewrite the equation in its standard form, the coefficients of \(y''\), \(y'\), and \(y\) behave nicely around this point. This ensures the ratio of these coefficients to \(x\), \(P(x)\) and \(Q(x)\), have either a limit or logarithmic singularity at \(x = 0\), thus making it feasible to solve using series methods.
Regular singular points are crucial as they guide us to use series expansion techniques that can neatly bypass problems typical at problematic points. This gives us a structured way to glean valuable solutions from seemingly tricky equations.

Indicial Equation

The Indicial Equation stands as the cornerstone for solving differential equations around a regular singular point using the Frobenius Method. It arises from substituting a series solution of the form \(y = \sum_{n=0}^{\infty} a_n x^{n+r}\) into the given differential equation. This way, the powers of \(x\) are balanced to produce clear coefficients. The Indicial Equation specifically deals with the lowest power of \(x\), which reveals insights into the behavior of solutions near the singular point.
In this example, after substitution and simplification, the Indicial Equation is found to be \(r^2 - r = 0\). When solved, it gives the roots \(r_1 = 0\) and \(r_2 = 1\). These roots signify the possible initial behavior of solutions near the singularity, and since they differ by an integer, we can anticipate that one solution will naturally be encompassed within the other.

Series Solution

The Series Solution is the approached method to solve differential equations at regular singular points, particularly using the Frobenius Method. After determining the indicial roots, we assume a solution \(y = \sum_{n=0}^{\infty} a_n x^{n+r}\) for each root. This method meticulously builds a solution step-by-step by equating the coefficients of like powers of \(x\) in the substituted series form of the equation.
This process leads to creating a recurrence relation that determines the sequence \(a_n\). For example, when \(r = 0\), assuming \(a_0 = 1\), we derive all other \(a_n\) through recurrence, eventually forming the first series solution \(y_1\). This systematic approach not only provides the finished form of one solution but serves as a springboard to derive any additional independent solutions necessary for constructing the general solution.

Second-order Differential Equation

Second-order differential equations, like \(x y'' + y' + y = 0\), play a fundamental role in mathematical modeling, describing numerous physical phenomena. These equations contain a second derivative, \(y''\), and are crucial for understanding dynamic systems.
Such equations are often challenging, particularly when singular points are involved. However, second-order equations offer more rich behavior in solutions, often leading to a general solution formed by two independent solutions, reflecting the equation's second-order nature. By exploring series solutions and the Indicial Equation, one gains entry to the complex world of such equations, revealing both challenges and structured techniques for solutions.