Question

Show that the given differential equation has a regular singular point at \(x=0 .\) Determine the indicial equation, the recurrence relation, and the roots of the indicial equation. Find the series solution \((x>0)\) corresponding to the larger root. If the roots are unequal and do not differ by an integer, find the series solution corresponding to the smaller root also. \(x^{2} y^{\prime \prime}-x(x+3) y^{\prime}+(x+3) y=0\)

Short answer

Question: Determine if the given differential equation has a regular singular point at x=0: \(x^2y'' - x(x+3)y' + (x+3)y = 0\) Answer: Yes, the given differential equation has a regular singular point at \(x=0\). The series solutions for the larger and smaller roots are: \[y_2(x) = a_0\left(1 - \frac{15}{5}x + \frac{6\cdot5}{5\cdot7}x^2 - \frac{7\cdot4}{5\cdot9}x^3 + \cdots\right)\] and \[y_1(x) = a_0\left(1 - \frac{3}{1}x + \frac{2\cdot(-1)}{1\cdot3}x^2 - \frac{1\cdot(-2)}{1\cdot5}x^3 + \cdots\right)\] where \(a_0\) is a constant.

Step-by-step solution

Write the equation in standard form

Divide both sides of the given equation by \(x^2\) to get: \[y'' - (x+3)y' + \frac{x+3}{x}y = 0\]

Check for a regular singular point at \(x=0\)

A point \(x_0\) is a regular singular point if both \((x-x_0)p(x)\) and \((x-x_0)^2q(x)\) are analytic at \(x=x_0\). In this case, \(p(x) = -(x+3)\) and \(q(x) = \frac{x+3}{x}\). Next, we compute: \[(x-0)p(x)= -x(x+3)\] \[(x-0)^2q(x) = x^2\frac{x+3}{x} = x(x+3)\] At \(x=0\), both of these expressions are well-defined, which means we have a regular singular point at \(x=0\).

Find the indicial equation

Assume a solution of the form \(y(x)=\sum_{n=0}^{\infty} a_n x^{r+n}\). Then \(y'(x)=\sum_{n=0}^{\infty} (r+n)a_n x^{r+n-1}\), and \(y''(x)=\sum_{n=0}^{\infty} (r+n)(r+n-1)a_n x^{r+n-2}\). Substitute these \(y\), \(y'\), and \(y''\) into the standard form of the differential equation and compare the coefficients of each power of \(x\). This gives the indicial equation: \[r(r-1) - 3r = 0\]

Find the roots of the indicial equation

The indicial equation factors as \(r(r-4) = 0\). Thus, the roots of the indicial equation are \(r_1 = 0\) and \(r_2 = 4\).

Find the recurrence relation

From the given equation, we derive the following relation for \(n>0\): \[((r+n)(r+n-1) - 3(r+n))a_n + (r+n+1)a_{n+1} = 0\] Rearranging and solving for \(a_{n+1}\): \[a_{n+1} = -\frac{((r+n)(r+n-1) - 3(r+n))}{r+n+1} a_n\]

Find the series solution for the larger root

Using the recurrence relation with the larger root \(r_2 = 4\) gives the series solution for \(x>0\): \[y_2(x) = a_0\left(1 - \frac{15}{5}x + \frac{6\cdot5}{5\cdot7}x^2 - \frac{7\cdot4}{5\cdot9}x^3 + \cdots\right)\] where \(a_0\) is a constant.

Find the series solution for the smaller root

Since the roots are unequal and do not differ by an integer, we can find the series solution for the smaller root \(r_1 = 0\) as well: \[y_1(x) = a_0\left(1 - \frac{3}{1}x + \frac{2\cdot(-1)}{1\cdot3}x^2 - \frac{1\cdot(-2)}{1\cdot5}x^3 + \cdots\right)\] where \(a_0\) is a constant. Now we have both series solutions corresponding to the larger and smaller roots.

Key concepts

Regular Singular Point

In differential equations, a singular point is a point where the equation of the form \(y'' + p(x) y' + q(x) y = 0\) becomes ill-behaved. However, a regular singular point is more manageable because it allows a certain type of solution technique. A point \(x = x_0\) is considered a regular singular point if both \((x-x_0)p(x)\) and \((x-x_0)^2q(x)\) have well-defined values as analytical functions around \(x_0\).
For the given equation \(x^2 y'' - x(x+3) y' + (x+3) y = 0\), simplifying the equation by dividing by \(x^2\), we express the components of the equation in standard form. Setting \(p(x) = -(x+3)\) and \(q(x) = \frac{x+3}{x}\), we evaluate if \(x=0\) is a regular singular point by checking:

• \((x-0)p(x) = -x(x+3)\), and
• \((x-0)^2q(x) = x(x+3)\).

Both these expressions are analytic at \(x=0\), confirming \(x=0\) as a regular singular point.
This characteristic is crucial as it underpins our ability to find series solutions.

Indicial Equation

The indicial equation emerges when you search for a series solution at a regular singular point. It is all about determining the values of \(r\) that would make the trial solution valid. To find the indicial equation, we assume a solution of the form \(y(x)=\sum_{n=0}^{\infty} a_n x^{r+n}\).
By substituting this trial solution along with its derivatives into the differential equation, we collect terms of like powers of \(x\). Setting the coefficient of the lowest power of \(x\) to zero gives us the indicial equation. In our case:
\[r(r-1) - 3r = 0\]
Solving this equation gives us the roots \(r_1 = 0\) and \(r_2 = 4\).
The roots tell us the possible exponents for the power series, guiding us in constructing the series solution corresponding to each root.

Series Solution

A series solution is a way to solve differential equations using a power series, especially around points termed as regular singular points. With the indicial equation's roots known, we begin constructing series solutions using these roots.
For the larger root \(r_2 = 4\), the recurrence relation provides a structured method for determining the coefficients \(a_n\). Applying the relation, we find the series solution for \(x > 0\):
\[ y_2(x) = a_0 \left( 1 - \frac{15}{5} x + \frac{6\cdot5}{5\cdot7} x^2 - \frac{7\cdot4}{5\cdot9} x^3 + \cdots \right) \]
The series solution reveals the distinct infinite sum structure characteristic of power series, capturing the behaviour of the differential equation around the chosen singular point.

Recurrence Relation

In the process of building the series solution, the recurrence relation plays an essential role. It tells us how to move from one coefficient of the series to the next, using the roots and the form of the differential equation.
After substituting the series form into the differential equation, we rearrange the terms to derive the recurrence relation:
\[ a_{n+1} = -\frac{((r+n)(r+n-1) - 3(r+n))}{(r+n+1)} a_n \]
This formula allows computation of subsequent terms \(a_{n+1}\) once an initial \(a_0\) is selected. For example, with larger root \(r_2 = 4\), setting the corresponding coefficients produces a coherent series solution. Similarly, for the smaller root \(r_1 = 0\), the recurrence relation permits deriving another series solution with a unique pattern. This mathematical construct simplifies the solution process for differential equations with regular singular points.