Test Questions on Regular Singular Points and Frobenius Series Solutions

📚 Quick Study Guide

• 🔍 A point $x_0$ is a singular point of the differential equation $P(x)y'' + Q(x)y' + R(x)y = 0$ if $P(x_0) = 0$.
• 💡 If $x_0$ is a singular point and the limits $\lim_{x \to x_0} (x-x_0)\frac{Q(x)}{P(x)}$ and $\lim_{x \to x_0} (x-x_0)^2\frac{R(x)}{P(x)}$ both exist and are finite, then $x_0$ is a regular singular point.
• 📝 The Frobenius method is used to find series solutions about a regular singular point. We assume a solution of the form $y(x) = \sum_{n=0}^{\infty} a_n (x-x_0)^{n+r}$, where $r$ is a constant to be determined.
• ➗ Substituting the Frobenius series into the differential equation and solving for the coefficients $a_n$ and the exponent $r$ (indicial roots) yields the Frobenius series solutions.
• 📌 The indicial equation is obtained by setting the coefficient of the lowest power of $(x-x_0)$ to zero. The roots $r_1$ and $r_2$ of the indicial equation determine the form of the two linearly independent solutions.
• 📈 If $r_1 - r_2$ is not an integer, then the two linearly independent solutions have the form $y_1(x) = \sum_{n=0}^{\infty} a_n (x-x_0)^{n+r_1}$ and $y_2(x) = \sum_{n=0}^{\infty} b_n (x-x_0)^{n+r_2}$.
• 🧪 If $r_1 = r_2$, or $r_1 - r_2$ is a positive integer, the second linearly independent solution may have a logarithmic term: $y_2(x) = y_1(x) \ln(x-x_0) + \sum_{n=0}^{\infty} c_n (x-x_0)^{n+r_2}$.

Practice Quiz

1. Question 1: Which of the following is the condition for $x_0$ to be a regular singular point of the differential equation $P(x)y'' + Q(x)y' + R(x)y = 0$?
   1. A) $P(x_0) = 0$, and $\lim_{x \to x_0} (x-x_0)\frac{Q(x)}{P(x)}$ and $\lim_{x \to x_0} (x-x_0)^2\frac{R(x)}{P(x)}$ both exist and are finite.
   2. B) $P(x_0) \neq 0$
   3. C) $Q(x_0) = 0$
   4. D) $R(x_0) = 0$
2. Question 2: What is the form of the Frobenius series solution around a regular singular point $x_0$?
   1. A) $y(x) = \sum_{n=0}^{\infty} a_n (x-x_0)^n$
   2. B) $y(x) = \sum_{n=0}^{\infty} a_n (x-x_0)^{n+r}$
   3. C) $y(x) = \sum_{n=0}^{\infty} a_n x^n$
   4. D) $y(x) = \sum_{n=0}^{\infty} a_n x^{n+r}$
3. Question 3: What is the purpose of the indicial equation in the Frobenius method?
   1. A) To find the coefficients $a_n$
   2. B) To determine the exponent $r$
   3. C) To simplify the differential equation
   4. D) To find the particular solution
4. Question 4: If the roots $r_1$ and $r_2$ of the indicial equation are such that $r_1 - r_2$ is not an integer, what is the form of the two linearly independent solutions?
   1. A) $y_1(x) = \sum_{n=0}^{\infty} a_n (x-x_0)^{n+r_1}$ and $y_2(x) = \sum_{n=0}^{\infty} b_n (x-x_0)^{n+r_2}$
   2. B) $y_1(x) = \sum_{n=0}^{\infty} a_n (x-x_0)^{n+r_1}$ and $y_2(x) = y_1(x) \ln(x-x_0)$
   3. C) $y_1(x) = \sum_{n=0}^{\infty} a_n (x-x_0)^{n+r_1}$ and $y_2(x) = x^{r_2}$
   4. D) $y_1(x) = x^{r_1}$ and $y_2(x) = x^{r_2}$
5. Question 5: What happens if $r_1 = r_2$?
   1. A) The second solution is simply a multiple of the first solution.
   2. B) The second solution involves a logarithmic term.
   3. C) There are no solutions.
   4. D) The Frobenius method fails.
6. Question 6: The differential equation $x^2y'' + xy' + (x^2 - \frac{1}{4})y = 0$ has a regular singular point at $x=0$. What are the roots of the indicial equation?
   1. A) $r = \pm \frac{1}{2}$
   2. B) $r = \pm 1$
   3. C) $r = 0, 1$
   4. D) $r = 0, -1$
7. Question 7: For the equation $(x^2 - 1)y'' + (x-1)y' + y = 0$, identify the regular singular points.
   1. A) $x = 1$ and $x = -1$
   2. B) $x = 1$ only
   3. C) $x = -1$ only
   4. D) No regular singular points