Irregular Singular Point Examples with Detailed Solutions for DE

📚 Quick Study Guide

• 🔍 Singular Point: A point $x_0$ is a singular point of the differential equation $P(x)y'' + Q(x)y' + R(x)y = 0$ if $P(x_0) = 0$.
• ➗ Regular Singular Point: A singular point $x_0$ is regular if $(x-x_0)\frac{Q(x)}{P(x)}$ and $(x-x_0)^2\frac{R(x)}{P(x)}$ are analytic at $x_0$. Otherwise, it is an irregular singular point.
• 💡 Irregular Singular Point: If either $(x-x_0)\frac{Q(x)}{P(x)}$ or $(x-x_0)^2\frac{R(x)}{P(x)}$ is not analytic at $x_0$, then $x_0$ is an irregular singular point.
• 📝 Frobenius Method: The Frobenius method is used to find series solutions near regular singular points. It may not work for irregular singular points.
• 📈 Identifying Irregular Points: Check if the limits $\lim_{x \to x_0} (x-x_0)\frac{Q(x)}{P(x)}$ and $\lim_{x \to x_0} (x-x_0)^2\frac{R(x)}{P(x)}$ exist and are finite. If either limit does not exist or is infinite, the point is irregular.

🧪 Practice Quiz

1. Question 1: For the equation $x^2y'' + xy' + (x^2-0.25)y = 0$, what type of singular point is $x=0$?
1. A) Regular Singular Point
2. B) Irregular Singular Point
3. C) Ordinary Point
4. D) Not a Singular Point
2. Question 2: Consider the equation $x^3y'' + 2x^2y' + y = 0$. What is the nature of the singular point at $x=0$?
1. A) Regular Singular Point
2. B) Irregular Singular Point
3. C) Ordinary Point
4. D) No Singular Point
3. Question 3: Which of the following equations has an irregular singular point at $x=0$?
1. A) $x^2y'' + xy' + y = 0$
2. B) $x^3y'' + xy' + y = 0$
3. C) $x^2y'' + x^2y' + y = 0$
4. D) $x^2y'' + xy' + x y = 0$
4. Question 4: For the equation $(x-2)y'' + y' + (x-2)y = 0$, what type of point is $x=2$?
1. A) Regular Singular Point
2. B) Irregular Singular Point
3. C) Ordinary Point
4. D) Not a Singular Point
5. Question 5: Examine the equation $x^4y'' + 2x^3y' + y = 0$. Determine the nature of the point at $x=0$.
1. A) Regular Singular Point
2. B) Irregular Singular Point
3. C) Ordinary Point
4. D) Not a Singular Point
6. Question 6: Which of the following differential equations has an irregular singular point at $x = 0$?
1. A) $x^2 y'' + x y' + (x^2 - 1/4) y = 0$
2. B) $x y'' + y' + y = 0$
3. C) $x^3 y'' + x y' + y = 0$
4. D) $x^2 y'' + x y' - 4 y = 0$
7. Question 7: What condition must be met for a singular point to be classified as regular, rather than irregular, at $x=x_0$?
1. A) $(x-x_0)P(x)/Q(x)$ and $(x-x_0)^2P(x)/R(x)$ are analytic at $x_0$
2. B) $(x-x_0)Q(x)/P(x)$ and $(x-x_0)^2R(x)/P(x)$ are analytic at $x_0$
3. C) $(x-x_0)R(x)/P(x)$ and $(x-x_0)^2Q(x)/P(x)$ are analytic at $x_0$
4. D) $P(x)$, $Q(x)$, and $R(x)$ are all analytic at $x_0$