Detailed Examples of Applying Euler's Formula to ODE Characteristic Equations

📚 Quick Study Guide

🔍 Euler's formula connects complex exponentials with trigonometric functions: $e^{ix} = \cos(x) + i\sin(x)$. 💡 For a characteristic equation resulting from an ODE, if you get complex roots of the form $\alpha \pm i\beta$, Euler's formula helps derive the real-valued solutions. 📝 The general solution corresponding to complex roots $\alpha \pm i\beta$ in the characteristic equation is $e^{\alpha x}(c_1\cos(\beta x) + c_2\sin(\beta x))$, where $c_1$ and $c_2$ are constants. 🧮 To apply Euler's formula to ODEs, first solve the characteristic equation. Then, if the roots are complex, use the formula to find the real-valued general solution. 📌Remember to match initial conditions (if given) to find the particular solution.

🧪 Practice Quiz

1. What is Euler's formula?
   1. A. $e^x = \cos(x) + i\sin(x)$
   2. B. $e^{ix} = \cos(x) + i\sin(x)$
   3. C. $e^{-ix} = \cos(x) - i\sin(x)$
   4. D. $e^{ix} = \sin(x) + i\cos(x)$
2. The characteristic equation of an ODE has roots $2 \pm 3i$. What is the form of the general solution?
   1. A. $e^{2x}(c_1\cos(3x) + c_2\sin(3x))$
   2. B. $e^{3x}(c_1\cos(2x) + c_2\sin(2x))$
   3. C. $c_1\cos(3x) + c_2\sin(3x)$
   4. D. $e^{-2x}(c_1\cos(3x) + c_2\sin(3x))$
3. If the characteristic equation has roots $-1 \pm i$, what is the general solution?
   1. A. $e^{x}(c_1\cos(x) + c_2\sin(x))$
   2. B. $e^{-x}(c_1\cos(x) + c_2\sin(x))$
   3. C. $c_1\cos(x) + c_2\sin(x)$
   4. D. $e^{-x}(c_1\cos(-x) + c_2\sin(-x))$
4. The roots of a characteristic equation are $i$ and $-i$. What is the general solution?
   1. A. $c_1\cos(x) + c_2\sin(x)$
   2. B. $e^{x}(c_1\cos(x) + c_2\sin(x))$
   3. C. $c_1e^{ix} + c_2e^{-ix}$
   4. D. $c_1\cos(ix) + c_2\sin(ix)$
5. Consider an ODE with characteristic equation roots of $0 \pm 2i$. Which of the following is the correct general solution?
   1. A. $c_1\cos(2x) + c_2\sin(2x)$
   2. B. $e^{2x}(c_1\cos(x) + c_2\sin(x))$
   3. C. $c_1e^{2ix} + c_2e^{-2ix}$
   4. D. $e^{0}(c_1\cos(2x) + c_2\sin(2x))$
6. What part of the complex root $\alpha \pm i\beta$ affects the amplitude of the solution?
   1. A. $\alpha$
   2. B. $\beta$
   3. C. $i$
   4. D. Both $\alpha$ and $\beta$
7. Given the characteristic equation $(r-2)^2 + 9 = 0$, what are the roots?
   1. A. $2 \pm 3$
   2. B. $2 \pm 9i$
   3. C. $2 \pm 3i$
   4. D. $-2 \pm 3i$