Question

Show that the given differential equation has a regular singular point at \(x=0,\) and determine two linearly independent solutions for \(x>0 .\) $$ x^{2} y^{\prime \prime}+4 x y^{\prime}+(2+x) y=0 $$

Short answer

Question: Determine whether the given point \(x=0\) is a regular singular point of the differential equation \(x^{2} y^{\prime \prime}+4 x y^{\prime}+(2+x) y=0\) and find two linearly independent solutions for \(x>0\). Answer: The given point \(x=0\) is a regular singular point of the differential equation. Two linearly independent solutions for \(x>0\) are given by \(y_1(x) = \sum_{n=0}^{\infty} c_n x^n\) and \(y_2(x) = x^{-3}\sum_{n=0}^{\infty} d_n x^n\), where \(c_n\) and \(d_n\) are coefficients determined by the Frobenius method.

Step-by-step solution

Identify the type of the differential equation

The given differential equation is $$ x^{2} y^{\prime \prime}+4 x y^{\prime}+(2+x) y=0. $$ This is a second-order linear ordinary differential equation.

Recognize singular points

For a second-order linear ordinary differential equation of the form $$ x^2y''+axy'+by=0, $$ the singular points are where the coefficients become infinite or the leading coefficient becomes zero. In this case, the only possible singular point is \(x=0\), as the leading coefficient \(x^2\) becomes zero.

Check if the singular point is regular

A singular point is regular if the coefficients of the transformed equation: $$ P(x) = \frac{a}{x} \text{ and } Q(x) = \frac{b}{x^2}, $$ are analytic functions at the given singular point. In our case, the given equation has coefficients \(a=4\) and \(b=2+x\). Thus, we have $$ P(x) = \frac{4}{x} \text{ and } Q(x) = \frac{2+x}{x^2}. $$ Both \(P(x)\) and \(Q(x)\) have simple poles at \(x=0\), so they are analytic at \(x=0\). Thus, the singular point \(x=0\) is a regular singular point.

Use Frobenius method to find linearly independent solutions

Since we have a regular singular point at \(x=0\), we can use the Frobenius method to find two linearly independent solutions for \(x > 0\). Starting with the ansatz $$ y(x) = x^r\sum_{n=0}^{\infty} c_n x^n, $$ we plug it into the differential equation, combine like terms, and solve for the indicial equation. Doing this gives the indicial equation: $$ r(r-1) + 4r + 2 = 0. $$ Solving this quadratic equation for \(r\), we get \(r_1 = 0\) and \(r_2 = -3\). Thus, we have two linearly independent solutions for \(x > 0\) given by $$ y_1(x) = \sum_{n=0}^{\infty} c_n x^n \text{ and } y_2(x) = x^{-3}\sum_{n=0}^{\infty} d_n x^n, $$ where \(c_n\) and \(d_n\) are the coefficients determined by the recurrence relation from the Frobenius method.

Key concepts

Frobenius method

The Frobenius method is a powerful technique used to find series solutions to second-order linear ordinary differential equations, especially when these equations have regular singular points. It involves expressing the solution in terms of power series, which helps in solving cases where traditional methods like reduction of order or characteristic equations fail.

In this method, the solution of a differential equation is sought in the form of a power series:

• \[ y(x) = x^r\sum_{n=0}^{\infty} c_n x^n \] where \( r \) is determined by the equation and \( c_n \) are the coefficients.
• This series is not only about ordinary points but starts with an initial factor \( x^r \). This accommodates the behavior of solutions around a singular point.

To apply the Frobenius method, you substitute the series form into the differential equation, equate coefficients of powers of \( x \), and solve for the indicial equation. Solving this gives the values of \( r \), which are crucial to form the series solution.

Second-order linear ordinary differential equation

A second-order linear ordinary differential equation (ODE) is a type of differential equation characterized by having the highest derivative as the second derivative. These equations often appear in physics and engineering contexts, delineating various phenomena like oscillations and wave propagation.

The general form of a second-order linear ODE is given by:

• \[ a(x) y'' + b(x) y' + c(x) y = 0 \] where \( a(x), b(x), \) and \( c(x) \) are functions of \( x \).
• The ordering refers to the highest derivative, which in this case is the second derivative \( y'' \).
• This type of ODE is called "linear" because the function \( y \) and its derivatives are not raised to any power greater than one.

Understanding the nature of the coefficients and the resulting behavior helps in selecting appropriate methods for solving these equations, such as the Frobenius method when dealing with regular singular points.

Indicial equation

The indicial equation is a significant component in the Frobenius method, critical for determining the values of the exponent \( r \) in the assumed series solution. This equation forms the basis for realizing the nature of solutions around a regular singular point.

When employing the Frobenius method, the indicial equation arises by substituting the Frobenius series into the differential equation and organizing coefficients of the lowest power of \( x \) into a separate equation:

• It results from the terms where \( x^r \) cannot be balanced out by terms involving \( x^{r+n} \). This isolates a quadratic or polynomial in \( r \) that provides possible values of \( r \).
• Solving the indicial equation usually involves finding roots \( r_1 \) and \( r_2 \), leading to two distinct solutions.

These values of \( r \) tell us about the types of solutions expected. Depending on whether the roots are distinct, equal, or complex, the nature of the solution changes. Hence, the indicial equation is pivotal for constructing complete solutions around a regular singular point.