Question

Consider the differential equation $$ y^{\prime \prime}+\frac{\alpha}{x^{s}} y^{\prime}+\frac{\beta}{x^{t}} y=0 $$ where \(\alpha \neq 0\) and \(\beta \neq 0\) are real numbers, and \(s\) and \(t\) are positive integers that for the moment are arbitrary. (a) Show that if \(s>1\) or \(t>2,\) then the point \(x=0\) is an irregular singular point. (b) Try to find a solution of Eq. (i) of the form $$ y=\sum_{n=0}^{\infty} a_{n} x^{r+n}, \quad x>0 $$ Show that if \(s=2\) and \(t=2,\) then there is only one possible value of \(r\) for which there is a formal solution of Eq. (i) of the form (ii). (c) Show that if \(\beta / \alpha=-1,0,1,2, \ldots,\) then the formal solution terminates and therefore is an actual solution. For other values of \(\beta / \alpha\) show that the formal series solution has a zero radius of convergence, and so does not represent an actual solution in any interval.

Short answer

Question: Determine if the point \(x=0\) is an irregular singular point for the following differential equation, and find the possible value of \(r\). $$x^{t-2}y'' + \frac{\alpha}{x^{s-2}}y' + \frac{\beta}{x^{t}}y = 0$$ with \(s>1\) and \(t>2\). Answer: When \(s>1\) or \(t>2\), the point \(x=0\) is an irregular singular point. The possible value of \(r\) is determined by: $$r = \pm\sqrt{\beta/\alpha}.$$

Step-by-step solution

(a) Singular Point Analysis

We will first check if the given point \(x=0\) is a singular point or not. To do this, we can rewrite the given equation in the standard form by dividing by \(x^{t-2}\). This yields $$ x^{t-2}y'' + \frac{\alpha}{x^{s-2}}y' + \beta y = 0. $$ Since \(x=0\) makes the coefficient of \(y'\) infinite if \(s > 1\) or the coefficient of \(y''\) infinite if \(t > 2\), we can conclude that \(x=0\) is an irregular singular point.

(b) Finding the Solution

Now we need to find a solution of the form $$ y = \sum_{n=0}^{\infty} a_n x^{r+n}, \quad x > 0. $$ To find the solution, we will substitute this series into the differential equation and then equate the terms of the same powers of \(x\). First, let's find the first and second derivatives of \(y\): $$ y' = \sum_{n=0}^{\infty} (r+n) a_n x^{r+n-1}, \quad y'' = \sum_{n=0}^{\infty} (r+n)(r+n-1) a_n x^{r+n-2}. $$ Now, substituting \(y, y'\), and \(y''\) into the differential equation: $$ \left[\sum_{n=0}^{\infty} (r+n)(r+n-1) a_n x^{r+n-2}\right] + \frac{\alpha}{x^{s}} \left[\sum_{n=0}^{\infty} (r+n) a_n x^{r+n-1}\right] + \frac{\beta}{x^{t}} \left[\sum_{n=0}^{\infty} a_n x^{r+n}\right] = 0 $$ We are given that \(s=2\) and \(t=2\), so we can simplify the equation as follows: $$ \left[\sum_{n=0}^{\infty} (r+n)(r+n-1) a_n x^{r+n-2}\right] + \frac{\alpha}{x^2} \left[\sum_{n=0}^{\infty} (r+n) a_n x^{r+n-1}\right] + \frac{\beta}{x^2} \left[\sum_{n=0}^{\infty} a_n x^{r+n}\right] = 0 $$ Now we consider the coefficients of the powers of \(x\) on both sides. For a non-trivial solution, we must have a non-zero solution for \(a_0\). By considering the coefficients of \(x^{r-2},x^{r-1},x^r\), we can find a recursion relation involving \(r\): $$ (r^2 - r) a_0 + ra_1 = 0 \\ r(r-1) a_0 + \alpha (r+1) a_1 = 0 \\ r(r+1) a_1 + \beta a_2 = 0 $$ Since \(a_0 \neq 0\), the last two equations combined reduce to: $$ r = \pm\sqrt{\beta/\alpha}. $$ Hence, there is only one possible value of \(r\) for which there is a formal solution.

(c) Solution Termination and Convergence

We can use the recursion relation to determine when the formal solution terminates and when it has a zero radius of convergence. For the series solution to terminate, we must have some positive integer \(p\) where \(a_p = 0\) and all following coefficients are also zero. By the recursion relation, this means that \(\beta / \alpha = -1,0,1,2,\ldots\). In these cases, the formal solution converges, leading to an actual solution. For all other values of \(\beta / \alpha\), the series solution does not terminate and continues infinitely. Therefore, the formal solution has a zero radius of convergence, so there is no actual solution for these values.

Key concepts

Singular Points

Singular points in differential equations are specific values of the variable at which the behavior of the solution changes dramatically.
This happens because the coefficients of the differential equation become undefined or infinite at these points. In the given differential equation, we identify singular points by setting up the equation such that it no longer makes sense, typically by observing when terms go towards infinity.
In our exercise, by altering the equation to the standard form, it becomes evident that the coefficients become infinite when plugging in specific values. This helps us determine if a singular point is regular or irregular:

• A point is a regular singular point if the coefficients go to infinity at a rate that still enables a solution around that point.
• A point is an irregular singular point if the coefficients' behavior makes it harder or impossible to find nearby solutions. For example, when either \(s > 1\) or \(t > 2\), \(x=0\) becomes an irregular singular point because the coefficients of \(y'\) or \(y''\) blow up in a manner that disrupts such solutions.

Understanding singular points is crucial for determining the nature of the solution we can achieve around the point of interest.

Series Solution

The series solution method involves expressing the solution to a differential equation as an infinite sum of terms.
Each term consists of a coefficient multiplied by a power of the independent variable.In the provided differential equation, the series solution involves setting up:\[y = \sum_{n=0}^{\infty} a_n x^{r+n}\]where \(a_n\) are coefficients and \(r\) is a parameter to be determined. Substituting this expression into the original differential equation allows us to align terms of similar powers of \(x\).
This alignment leads to a system of equations that we solve sequentially to find the coefficients \(a_n\) and the parameter \(r\).The primary goal of this method is to analyze all powers of \(x\), creating a series expression that satisfies the differential equation for all terms. This helps to confirm the consistency and correctness of the proposed series as a potential solution.
If the series does not converge or comply properly within the desired framework, it could point to deeper issues or different types of singular points present in the equation.

Formal and Actual Solutions

Formal solutions refer to theoretically computed solutions using a series expansion method.
These solutions don't necessarily imply the possibility of practical or physically meaningful solutions.In our case, finding a formal solution involves determining conditions under which the series ends neatly or diverges infinitely:

• If the series terminates, it means that the formal solution matches the actual solution of the differential equation, providing a realistic and applicable solution that fits a given initial condition.
• If the calculated formal series solution does not terminate, it continues indefinitely and lacks valid convergence in any interval, indicating no actual solution is represented by the series.

In this scenario, certain ratios of parameters, like \(\beta/\alpha = -1, 0, 1, 2, \ldots\), lead to a terminated series, ensuring that a formal solution successfully represents an actual solution.
Variations in these specific ratios result in non-terminating series, limiting the practical utility of the derived solutions.