Practice quiz on Case 3 Frobenius solutions with logarithmic terms.

📚 Topic Summary

When solving second-order linear differential equations using the Frobenius method, Case 3 arises when the roots $r_1$ and $r_2$ of the indicial equation are equal. In this situation, we obtain one Frobenius solution directly. A second, linearly independent solution involves a logarithmic term. The general form of the second solution often involves the first solution multiplied by $\ln(x)$, along with another Frobenius series. Careful calculation of coefficients is necessary to determine this second solution.

🧠 Part A: Vocabulary

Match each term with its definition:

1. Term: Indicial Equation
2. Term: Frobenius Method
3. Term: Logarithmic Term
4. Term: Linearly Independent
5. Term: Second-Order Linear Differential Equation

1. Definition: A term involving the natural logarithm, often arising in solutions when roots of the indicial equation are repeated.
2. Definition: An equation of the form $a(x)y'' + b(x)y' + c(x)y = 0$ where $a(x)$, $b(x)$, and $c(x)$ are functions of $x$.
3. Definition: A method for finding series solutions to differential equations, particularly near regular singular points.
4. Definition: A quadratic equation obtained from the recurrence relation in the Frobenius method, used to determine the possible values of the exponent.
5. Definition: Two solutions where neither is a constant multiple of the other; they contribute unique information to the general solution.

📝 Part B: Fill in the Blanks

In Case 3 of the Frobenius method, when the roots of the indicial equation are _____, the second solution will contain a _____ term. This term arises because the standard Frobenius series method only yields _____ linearly independent solution in this case. The general form of the second solution involves the first solution multiplied by _____, plus another _____ series.

💡 Part C: Critical Thinking

Explain why a logarithmic term appears in the second linearly independent solution for Case 3 of the Frobenius method. What does its presence tell us about the behavior of the solutions near the singular point?