Illustrative Examples of Higher-Order Linear Homogeneous Equations with Constant Coefficients

📚 Quick Study Guide

• 🔢 A linear homogeneous differential equation of order $n$ has the form: $a_n y^{(n)} + a_{n-1} y^{(n-1)} + ... + a_1 y' + a_0 y = 0$, where $a_i$ are constants.
• 🔑 To solve, assume a solution of the form $y = e^{rx}$, where $r$ is a constant.
• 📝 Substitute $y = e^{rx}$ into the differential equation and solve for $r$. This results in the characteristic equation: $a_n r^n + a_{n-1} r^{n-1} + ... + a_1 r + a_0 = 0$.
• ➗ The roots of the characteristic equation determine the form of the general solution.
• 💡 Distinct real roots $r_1, r_2, ..., r_n$ give the general solution: $y = c_1 e^{r_1 x} + c_2 e^{r_2 x} + ... + c_n e^{r_n x}$.
• 📈 Repeated real root $r$ (with multiplicity $k$) gives the solutions: $e^{rx}, xe^{rx}, x^2 e^{rx}, ..., x^{k-1} e^{rx}$.
• 🧭 Complex conjugate roots $\alpha \pm i\beta$ give the solutions: $e^{\alpha x} \cos(\beta x)$ and $e^{\alpha x} \sin(\beta x)$.

Practice Quiz

1. Question 1: What is the characteristic equation for the differential equation $y'' + 5y' + 6y = 0$?
1. A) $r + 5 = 0$
2. B) $r^2 + 5r + 6 = 0$
3. C) $r^2 + 6 = 0$
4. D) $5r + 6 = 0$
2. Question 2: Given the characteristic equation $r^2 - 4r + 4 = 0$, what is the general solution of the corresponding differential equation?
1. A) $y = c_1 e^{2x}$
2. B) $y = c_1 e^{2x} + c_2 x e^{2x}$
3. C) $y = c_1 e^{2x} + c_2 e^{-2x}$
4. D) $y = c_1 \cos(2x) + c_2 \sin(2x)$
3. Question 3: What is the general solution for the differential equation $y''' - 6y'' + 11y' - 6y = 0$, given the roots of the characteristic equation are $r = 1, 2, 3$?
1. A) $y = c_1 e^x + c_2 e^{2x} + c_3 e^{3x}$
2. B) $y = c_1 e^{-x} + c_2 e^{-2x} + c_3 e^{-3x}$
3. C) $y = c_1 e^x + c_2 x e^x + c_3 x^2 e^x$
4. D) $y = c_1 \cos(x) + c_2 \sin(2x) + c_3 e^{3x}$
4. Question 4: Which of the following is a solution to $y'' + 4y = 0$?
1. A) $e^{2x}$
2. B) $\sin(2x)$
3. C) $x^2$
4. D) $e^{-2x}$
5. Question 5: If the characteristic equation has roots $r = 2 \pm 3i$, what is the general solution?
1. A) $y = c_1 e^{2x} \cos(3x) + c_2 e^{2x} \sin(3x)$
2. B) $y = c_1 e^{2x} + c_2 e^{3x}$
3. C) $y = c_1 \cos(2x) + c_2 \sin(3x)$
4. D) $y = c_1 e^{-2x} \cos(3x) + c_2 e^{-2x} \sin(3x)$
6. Question 6: What form of solution should you assume for a differential equation with a repeated root $r=5$ (multiplicity 3)?
1. A) $c_1e^{5x}$
2. B) $c_1e^{5x} + c_2xe^{5x}$
3. C) $c_1e^{5x} + c_2xe^{5x} + c_3x^2e^{5x}$
4. D) $c_1e^{5x} + c_2e^{-5x} + c_3e^{0x}$
7. Question 7: For the equation $y'''-y''+y'-y=0$, one root is $r=1$. What is a possible solution to the differential equation?
1. A) $e^x$
2. B) $\sin(x)$
3. C) $\cos(x)$
4. D) All of the above