Solved Examples: Distinct Real Roots for Homogeneous Linear DEs

📚 Quick Study Guide

• 🔢 Consider a homogeneous linear differential equation of the form: $a_n y^{(n)} + a_{n-1} y^{(n-1)} + ... + a_1 y' + a_0 y = 0$, where $a_i$ are constants.
• 🔑 Assume a solution of the form $y = e^{rx}$. Substituting this into the differential equation yields the characteristic equation: $a_n r^n + a_{n-1} r^{n-1} + ... + a_1 r + a_0 = 0$.
• 🌱 If the characteristic equation has $n$ distinct real roots $r_1, r_2, ..., r_n$, then the general solution is given by: $y(x) = c_1 e^{r_1 x} + c_2 e^{r_2 x} + ... + c_n e^{r_n x}$, where $c_i$ are arbitrary constants.
• ✍️ To find the particular solution, use the initial conditions to solve for the constants $c_1, c_2, ..., c_n$.
• 💡 Remember that distinct real roots mean each root is different (e.g., $r_1 \neq r_2 \neq ... \neq r_n$).

Practice Quiz

1. Which of the following is the correct form of the general solution for a second-order homogeneous linear differential equation with distinct real roots $r_1$ and $r_2$?
   1. $y(x) = c_1 e^{r_1 x} + c_2 e^{r_2 x}$
   2. $y(x) = (c_1 + c_2 x) e^{r_1 x}$
   3. $y(x) = c_1 \cos(r_1 x) + c_2 \sin(r_2 x)$
   4. $y(x) = c_1 e^{(r_1 + r_2)x}$
2. What is the characteristic equation for the differential equation $y'' - 5y' + 6y = 0$?
   1. $r^2 - 5r + 6 = 0$
   2. $r - 5r + 6 = 0$
   3. $r^2 + 5r + 6 = 0$
   4. $r^2 - 5r - 6 = 0$
3. The roots of the characteristic equation are $r_1 = 2$ and $r_2 = 3$. What is the general solution of the corresponding homogeneous linear differential equation?
   1. $y(x) = c_1 e^{2x} + c_2 e^{3x}$
   2. $y(x) = c_1 e^{2x} + c_2 x e^{2x}$
   3. $y(x) = c_1 e^{3x} + c_2 x e^{3x}$
   4. $y(x) = c_1 e^{5x}$
4. If the roots of the characteristic equation are $r = -1$ and $r = 4$, what is the differential equation?
   1. $y'' - 3y' - 4y = 0$
   2. $y'' + 3y' - 4y = 0$
   3. $y'' - 5y' + 4y = 0$
   4. $y'' + 5y' + 4y = 0$
5. Find the general solution of $y'' - y = 0$.
   1. $y(x) = c_1 e^{x} + c_2 e^{-x}$
   2. $y(x) = c_1 e^{x} + c_2 x e^{x}$
   3. $y(x) = c_1 \cos(x) + c_2 \sin(x)$
   4. $y(x) = c_1 e^{-x} + c_2 x e^{-x}$
6. Given the initial conditions $y(0) = 1$ and $y'(0) = 0$, find the particular solution for the differential equation $y'' - 4y = 0$.
   1. $y(x) = \frac{1}{2} e^{2x} + \frac{1}{2} e^{-2x}$
   2. $y(x) = e^{2x}$
   3. $y(x) = e^{-2x}$
   4. $y(x) = \cos(2x)$
7. What is the general solution to $y''' - 6y'' + 11y' - 6y = 0$?
   1. $y(x) = c_1 e^x + c_2 e^{2x} + c_3 e^{3x}$
   2. $y(x) = c_1 e^x + c_2 e^{2x}$
   3. $y(x) = c_1 e^{2x} + c_2 e^{3x}$
   4. $y(x) = c_1 e^x + c_2 xe^x + c_3 x^2 e^x$