Differential Equations Test Questions: The Wronskian and its applications

📚 Quick Study Guide

• 🔎Definition: The Wronskian of $n$ functions $f_1(x), f_2(x), ..., f_n(x)$ is the determinant of the matrix formed by the functions and their derivatives up to order $n-1$.
• ➗Formula: $W(f_1, f_2, ..., f_n)(x) = \begin{vmatrix} f_1(x) & f_2(x) & ... & f_n(x) \\ f_1'(x) & f_2'(x) & ... & f_n'(x) \\ ... & ... & ... & ... \\ f_1^{(n-1)}(x) & f_2^{(n-1)}(x) & ... & f_n^{(n-1)}(x) \end{vmatrix}$
• 🔑Linear Independence: If $W(f_1, f_2, ..., f_n)(x) \neq 0$ for some $x$ in the interval, then the functions are linearly independent.
• 📝Linear Dependence: If the functions are linearly dependent, then $W(f_1, f_2, ..., f_n)(x) = 0$ for all $x$ in the interval. The converse is not always true.
• 💡Abel's Theorem: For a second-order linear homogeneous differential equation $y'' + p(x)y' + q(x)y = 0$, the Wronskian of two solutions $y_1$ and $y_2$ satisfies $W(y_1, y_2)(x) = c \cdot e^{-\int p(x) dx}$, where $c$ is a constant.

Practice Quiz

1. Question 1: The Wronskian of two solutions to a second-order linear homogeneous differential equation is found to be identically zero. What can you conclude about the solutions?
   1. A. The solutions are linearly independent.
   2. B. The solutions are linearly dependent.
   3. C. The solutions may be linearly independent or dependent; more information is needed.
   4. D. The solutions are trivial.
2. Question 2: Calculate the Wronskian of the functions $f(x) = x$ and $g(x) = x^2$.
   1. A. $W(x, x^2) = x^2$
   2. B. $W(x, x^2) = x$
   3. C. $W(x, x^2) = 2x$
   4. D. $W(x, x^2) = 0$
3. Question 3: Given the differential equation $y'' + y = 0$, two solutions are $y_1 = \cos(x)$ and $y_2 = \sin(x)$. Calculate their Wronskian.
   1. A. $W(\cos(x), \sin(x)) = 0$
   2. B. $W(\cos(x), \sin(x)) = -1$
   3. C. $W(\cos(x), \sin(x)) = 1$
   4. D. $W(\cos(x), \sin(x)) = \cos(x) \sin(x)$
4. Question 4: If the Wronskian of two functions is non-zero at a point, what can be concluded about the functions at that point?
   1. A. They are linearly dependent.
   2. B. They are linearly independent.
   3. C. They are equal.
   4. D. One is the derivative of the other.
5. Question 5: According to Abel's Theorem, how does the Wronskian of two solutions to $y'' + p(x)y' + q(x)y = 0$ behave?
   1. A. It is always constant.
   2. B. It is always zero.
   3. C. It is proportional to $e^{\int p(x) dx}$.
   4. D. It is proportional to $e^{-\int p(x) dx}$.
6. Question 6: Determine if the functions $f(x) = e^x$ and $g(x) = 2e^x$ are linearly independent by computing their Wronskian.
   1. A. Linearly independent, $W \neq 0$
   2. B. Linearly dependent, $W = 0$
   3. C. Linearly independent, $W = 0$
   4. D. Linearly dependent, $W \neq 0$
7. Question 7: What is a limitation of using the Wronskian to determine linear independence?
   1. A. It only works for first-order differential equations.
   2. B. It can only determine linear dependence, not independence.
   3. C. If the Wronskian is zero, it doesn't guarantee linear dependence.
   4. D. It requires solving the differential equation first.