Test Questions: Solving Characteristic Equation Case 3 with Euler's Formula

📚 Quick Study Guide

• 🔢 When solving a second-order linear homogeneous differential equation with constant coefficients, $ay'' + by' + cy = 0$, Case 3 arises when the characteristic equation $ar^2 + br + c = 0$ has complex conjugate roots, $r = \alpha \pm i\beta$.
• ➗ Euler's formula states that $e^{i\theta} = \cos(\theta) + i\sin(\theta)$. This formula is crucial for expressing complex exponentials in terms of trigonometric functions.
• ✨ The general solution for Case 3 is given by $y(x) = e^{\alpha x}(c_1 \cos(\beta x) + c_2 \sin(\beta x))$, where $c_1$ and $c_2$ are arbitrary constants determined by initial conditions.
• 📐 $\alpha$ represents the real part of the complex root, and $\beta$ represents the imaginary part. These values directly influence the amplitude and frequency of the sinusoidal components in the solution.
• 📝 Remember to apply initial conditions (if provided) to find the specific values of $c_1$ and $c_2$, thus obtaining the particular solution to the differential equation.

Practice Quiz

1. Question 1: What type of roots does the characteristic equation have in Case 3?
   1. Real and distinct
   2. Real and repeated
   3. Complex conjugate
   4. Rational
2. Question 2: Given the complex root $r = 2 + 3i$, what are the values of $\alpha$ and $\beta$?
   1. $\alpha = 3$, $\beta = 2$
   2. $\alpha = 2$, $\beta = 3$
   3. $\alpha = -2$, $\beta = 3$
   4. $\alpha = 2$, $\beta = -3$
3. Question 3: What is the general form of the solution in Case 3?
   1. $y(x) = c_1e^{r_1x} + c_2e^{r_2x}$
   2. $y(x) = (c_1 + c_2x)e^{rx}$
   3. $y(x) = e^{\alpha x}(c_1 \cos(\beta x) + c_2 \sin(\beta x))$
   4. $y(x) = c_1 \cos(\beta x) + c_2 \sin(\beta x)$
4. Question 4: According to Euler's formula, $e^{i\theta}$ is equal to:
   1. $\sin(\theta) + i\cos(\theta)$
   2. $\cos(\theta) - i\sin(\theta)$
   3. $\cos(\theta) + i\sin(\theta)$
   4. $\sin(\theta) - i\cos(\theta)$
5. Question 5: If the characteristic equation is $r^2 + 4 = 0$, what are the roots?
   1. $r = \pm 2$
   2. $r = \pm 4$
   3. $r = \pm 2i$
   4. $r = \pm 4i$
6. Question 6: Given the roots $r = -1 \pm 2i$, what is the general solution?
   1. $y(x) = e^{-x}(c_1 \cos(2x) + c_2 \sin(2x))$
   2. $y(x) = e^{x}(c_1 \cos(2x) + c_2 \sin(2x))$
   3. $y(x) = c_1 \cos(-x) + c_2 \sin(2x)$
   4. $y(x) = e^{-x}(c_1 \cos(x) + c_2 \sin(2x))$
7. Question 7: In the general solution $y(x) = e^{\alpha x}(c_1 \cos(\beta x) + c_2 \sin(\beta x))$, what do $c_1$ and $c_2$ represent?
   1. The roots of the characteristic equation
   2. Arbitrary constants determined by initial conditions
   3. The real and imaginary parts of the complex root
   4. The coefficients of the differential equation