Test questions for homogeneous second-order linear differential equations.

📚 Quick Study Guide

🔍 Definition: A homogeneous second-order linear differential equation has the form $ay'' + by' + cy = 0$, where $a$, $b$, and $c$ are constants. 💡 Characteristic Equation: To solve such equations, we form the characteristic equation $ar^2 + br + c = 0$. 📝 Roots: The nature of the roots $r_1$ and $r_2$ of the characteristic equation determines the form of the general solution. ➗ Distinct Real Roots: If $r_1 \neq r_2$ are real, the general solution is $y(x) = c_1e^{r_1x} + c_2e^{r_2x}$. ➕ Repeated Real Roots: If $r_1 = r_2 = r$, the general solution is $y(x) = c_1e^{rx} + c_2xe^{rx}$. 🧭 Complex Conjugate Roots: If $r_{1,2} = \alpha \pm i\beta$, the general solution is $y(x) = e^{\alpha x}(c_1\cos(\beta x) + c_2\sin(\beta x))$. 📖 Initial Conditions: Initial conditions (e.g., $y(0) = y_0$, $y'(0) = y'_0$) are used to determine the constants $c_1$ and $c_2$.

Practice Quiz

1. What is the characteristic equation for the differential equation $2y'' - 5y' - 3y = 0$?
   1. $2r - 5r - 3 = 0$
   2. $2r^2 - 5r - 3 = 0$
   3. $2r^2 + 5r - 3 = 0$
   4. $2r^2 - 5r + 3 = 0$
2. The characteristic equation of a second-order linear homogeneous differential equation has roots $r_1 = 3$ and $r_2 = -2$. What is the general solution?
   1. $y(x) = c_1e^{3x} + c_2e^{-2x}$
   2. $y(x) = c_1e^{-3x} + c_2e^{2x}$
   3. $y(x) = c_1e^{3x} - c_2e^{-2x}$
   4. $y(x) = c_1e^{x} + c_2e^{-x}$
3. For the differential equation $y'' + 6y' + 9y = 0$, what type of roots does the characteristic equation have?
   1. Distinct real roots
   2. Repeated real roots
   3. Complex conjugate roots
   4. Imaginary roots
4. If the characteristic equation has complex roots $2 \pm 3i$, what is the general solution to the differential equation?
   1. $y(x) = e^{2x}(c_1\cos(3x) + c_2\sin(3x))$
   2. $y(x) = c_1\cos(2x) + c_2\sin(2x)$
   3. $y(x) = e^{3x}(c_1\cos(2x) + c_2\sin(2x))$
   4. $y(x) = e^{-2x}(c_1\cos(3x) + c_2\sin(3x))$
5. What is the form of the general solution if the characteristic equation has a repeated root of $r = -1$?
   1. $y(x) = c_1e^{-x}$
   2. $y(x) = c_1e^{-x} + c_2xe^{-x}$
   3. $y(x) = c_1e^{-x} + c_2e^{x}$
   4. $y(x) = c_1e^{x} + c_2xe^{x}$
6. Given the differential equation $y'' + 4y = 0$, what are the roots of the characteristic equation?
   1. $r = \pm 2$
   2. $r = \pm 2i$
   3. $r = 2, 0$
   4. $r = \pm 4$
7. Which of the following equations is a homogeneous second-order linear differential equation?
   1. $y'' + 3y' + 2y = x$
   2. $y'' + 3y' + 2y = 0$
   3. $y'' + 3y' + 2 = 0$
   4. $y' + 3y = 0$