Solved problems: Linear independence and Wronskian examples for 2nd order ODEs

📚 Quick Study Guide

• 🔢 Linear Independence: Two functions $f(x)$ and $g(x)$ are linearly independent on an interval if the equation $c_1f(x) + c_2g(x) = 0$ for all $x$ in the interval implies that $c_1 = c_2 = 0$. Otherwise, they are linearly dependent.
• 🤔 Wronskian: The Wronskian of two functions $f(x)$ and $g(x)$ is defined as: $$W(f, g)(x) = \begin{vmatrix} f(x) & g(x) \\ f'(x) & g'(x) \end{vmatrix} = f(x)g'(x) - f'(x)g(x)$$
• ❗ Using the Wronskian:
  • 🧪 If $W(f, g)(x) \neq 0$ for some $x$ in the interval, then $f$ and $g$ are linearly independent on that interval.
  • 📝 If $f$ and $g$ are solutions to a homogeneous linear 2nd order ODE, and $W(f, g)(x) = 0$ for all $x$ in the interval, then $f$ and $g$ are linearly dependent on that interval.
• 📈 General Solution: If $y_1(x)$ and $y_2(x)$ are linearly independent solutions of a homogeneous linear 2nd order ODE, then the general solution is $y(x) = c_1y_1(x) + c_2y_2(x)$, where $c_1$ and $c_2$ are arbitrary constants.

Practice Quiz

1. Which of the following pairs of functions are linearly independent?
   1. A) $f(x) = x, g(x) = 2x$
   2. B) $f(x) = sin(x), g(x) = 2sin(x)$
   3. C) $f(x) = e^x, g(x) = xe^x$
   4. D) $f(x) = x^2, g(x) = 3x^2$
2. Calculate the Wronskian of $f(x) = x$ and $g(x) = e^x$.
   1. A) $xe^x - e^x$
   2. B) $xe^x + e^x$
   3. C) $e^x - xe^x$
   4. D) $e^x$
3. If the Wronskian of two solutions to a 2nd order homogeneous linear ODE is zero everywhere on an interval, what can you conclude about the solutions?
   1. A) They are linearly independent.
   2. B) They are linearly dependent.
   3. C) One of the solutions is trivial.
   4. D) The ODE has no solution.
4. Given $y_1(x) = cos(x)$ and $y_2(x) = sin(x)$, find their Wronskian.
   1. A) $-1$
   2. B) $0$
   3. C) $1$
   4. D) $sin(x)cos(x)$
5. Which pair of functions is linearly dependent?
   1. A) $x, x^2$
   2. B) $e^x, e^{-x}$
   3. C) $sin^2(x), 1 - cos^2(x)$
   4. D) $ln(x), x$
6. Suppose $y_1(x)$ and $y_2(x)$ are solutions to $y'' + p(x)y' + q(x)y = 0$. If $W(y_1, y_2)(x) = e^{-x^2}$, are $y_1$ and $y_2$ linearly independent?
   1. A) Yes, they are linearly independent.
   2. B) No, they are linearly dependent.
   3. C) Cannot be determined.
   4. D) They are orthogonal.
7. What does a non-zero Wronskian imply for a set of solutions to a homogeneous linear differential equation?
   1. A) The solutions are linearly dependent.
   2. B) The solutions are linearly independent.
   3. C) The solutions are trivial.
   4. D) The solutions are complex.