D-operator solutions for higher-order homogeneous equations (Examples)

📚 Quick Study Guide

🔍 Understanding D-operator: The D-operator, denoted as $D$, represents the derivative operator, i.e., $Dy = \frac{dy}{dx}$, $D^2y = \frac{d^2y}{dx^2}$, and so on. 💡 Homogeneous Equations: A linear homogeneous differential equation of order $n$ has the form $a_n(x)y^{(n)} + a_{n-1}(x)y^{(n-1)} + ... + a_1(x)y' + a_0(x)y = 0$. 📝 Constant Coefficients: When the coefficients $a_i$ are constants, we can use the D-operator to solve these equations more efficiently. ➗ Auxiliary Equation: Replace $D$ with $m$ to obtain the auxiliary equation $a_nm^n + a_{n-1}m^{n-1} + ... + a_1m + a_0 = 0$. ➕ Distinct Real Roots: If the auxiliary equation has distinct real roots $m_1, m_2, ..., m_n$, the general solution is $y = c_1e^{m_1x} + c_2e^{m_2x} + ... + c_ne^{m_nx}$. 📈 Repeated Real Roots: If $m_1$ is a repeated root with multiplicity $k$, the corresponding part of the solution is $(c_1 + c_2x + ... + c_kx^{k-1})e^{m_1x}$. 🧭 Complex Roots: If the auxiliary equation has complex roots $α ± iβ$, the corresponding part of the solution is $e^{αx}(c_1cos(βx) + c_2sin(βx))$.

Practice Quiz

1. The D-operator is defined as: A. $Dy = \int y dx$ B. $Dy = \frac{dy}{dx}$ C. $Dy = y^2$ D. $Dy = e^y$ 2. Which of the following is a homogeneous linear differential equation with constant coefficients? A. $x^2y'' + xy' + y = 0$ B. $y'' + xy' + y = x$ C. $y'' + 3y' + 2y = 0$ D. $y'' + 3y' + 2y = e^x$ 3. What is the auxiliary equation for the differential equation $y'' - 5y' + 6y = 0$? A. $m - 5m + 6 = 0$ B. $m^2 - 5m + 6 = 0$ C. $m^2 + 5m + 6 = 0$ D. $m^2 - 5m = 0$ 4. If the auxiliary equation has roots $m_1 = 2$ and $m_2 = 3$, the general solution is: A. $y = c_1e^{2x} + c_2e^{3x}$ B. $y = c_1e^{-2x} + c_2e^{-3x}$ C. $y = c_1e^{2x} - c_2e^{3x}$ D. $y = c_1x^2 + c_2x^3$ 5. If the auxiliary equation has a repeated root $m = 1$ with multiplicity 2, the corresponding part of the solution is: A. $c_1e^x$ B. $c_1e^x + c_2e^x$ C. $(c_1 + c_2x)e^x$ D. $c_1e^x + c_2xe^{2x}$ 6. If the auxiliary equation has complex roots $2 ± 3i$, the corresponding part of the solution is: A. $e^{2x}(c_1cos(3x) + c_2sin(3x))$ B. $e^{3x}(c_1cos(2x) + c_2sin(2x))$ C. $c_1cos(2x) + c_2sin(3x)$ D. $e^{2x}(c_1sin(3x) + c_2cos(2x))$ 7. What is the solution to the differential equation $(D^2 + 4D + 4)y = 0$? A. $y = c_1e^{2x} + c_2e^{2x}$ B. $y = c_1e^{-2x} + c_2e^{-2x}$ C. $y = (c_1 + c_2x)e^{-2x}$ D. $y = (c_1 + c_2x)e^{2x}$

Click to see Answers

1. B
2. C
3. B
4. A
5. C
6. A
7. C