Solved Examples: Second-Order Linear ODEs with Real Distinct Roots

📚 Quick Study Guide

• 🔍 General Form: A second-order linear ODE is given by $ay'' + by' + cy = 0$, where $a$, $b$, and $c$ are constants.
• 📈 Characteristic Equation: To solve this, we form the characteristic equation: $ar^2 + br + c = 0$.
• ➗ Real Distinct Roots: If the characteristic equation has two distinct real roots, $r_1$ and $r_2$, the general solution is $y(x) = c_1e^{r_1x} + c_2e^{r_2x}$, where $c_1$ and $c_2$ are arbitrary constants.
• 💡 Finding Roots: Use the quadratic formula $r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ to find the roots of the characteristic equation.
• 📝 Applying Initial Conditions: If initial conditions $y(x_0) = y_0$ and $y'(x_0) = y'_0$ are given, use them to solve for $c_1$ and $c_2$.

Practice Quiz

1. Question 1: The general solution to $y'' - 3y' + 2y = 0$ is:
   1. $y(x) = c_1e^x + c_2e^{2x}$
   2. $y(x) = c_1e^{-x} + c_2e^{-2x}$
   3. $y(x) = c_1xe^x + c_2e^{2x}$
   4. $y(x) = c_1e^x + c_2xe^{2x}$
2. Question 2: The characteristic equation for $2y'' + 5y' - 3y = 0$ is:
   1. $2r^2 + 5r - 3 = 0$
   2. $2r^2 - 5r + 3 = 0$
   3. $2r + 5 = 0$
   4. $r^2 + 5r - 3 = 0$
3. Question 3: Given $y'' + y' - 6y = 0$, with $y(0) = 1$ and $y'(0) = 0$, find $y(x)$.
   1. $y(x) = \frac{3}{5}e^{2x} + \frac{2}{5}e^{-3x}$
   2. $y(x) = \frac{2}{5}e^{2x} + \frac{3}{5}e^{-3x}$
   3. $y(x) = e^{2x} + e^{-3x}$
   4. $y(x) = e^{-2x} + e^{3x}$
4. Question 4: The roots of the characteristic equation for $y'' - 4y' + 3y = 0$ are:
   1. $r = 1, 3$
   2. $r = -1, -3$
   3. $r = 1, -3$
   4. $r = -1, 3$
5. Question 5: Which of the following is a second-order linear ODE with constant coefficients?
   1. $x^2y'' + xy' + y = 0$
   2. $y'' + xy' + y = x$
   3. $y'' + 3y' + 2y = 0$
   4. $y'' + y' + xy = 0$
6. Question 6: The general solution of $y'' - y = 0$ is:
   1. $y(x) = c_1e^x + c_2e^{-x}$
   2. $y(x) = c_1\cos(x) + c_2\sin(x)$
   3. $y(x) = c_1e^x + c_2x$
   4. $y(x) = c_1 + c_2e^{-x}$
7. Question 7: For the ODE $y'' + 2y' - 8y = 0$, the roots of the characteristic equation are:
   1. $r = 2, -4$
   2. $r = -2, 4$
   3. $r = -2, -4$
   4. $r = 2, 4$