1) Introduction: Beyond First-Order Equations 📚
This topic extends our knowledge of differential equations to higher-order differential equations and systems of ordinary differential equations (ODEs). These are crucial for modeling physical systems like oscillations, electrical circuits, and population dynamics. We’ll explore methods to solve linear higher-order ODEs with constant coefficients and systems using eigenvalues and eigenvectors.
We’ll cover:
- Higher-Order ODEs: Solving with characteristic equations.
- Systems of ODEs: Matrix methods and phase planes.
- Applications: Real-world examples like damped motion.
Quick Recap: First-order ODEs used separable methods; now we handle multiple derivatives and coupled equations.
2) Higher-Order Differential Equations 🎓
A higher-order ODE is of the form \( a_n y^{(n)} + \cdots + a_1 y' + a_0 y = f(x) \).
Definition 28.1: Higher-Order ODE
For a linear ODE with constant coefficients, solve the characteristic equation \( a_n r^n + \cdots + a_1 r + a_0 = 0 \). Roots determine the general solution.
Example 1: Second-Order Homogeneous
Solve \( y'' + 4y = 0 \).
- Characteristic: \( r^2 + 4 = 0 \), \( r = \pm 2i \).
- Solution: \( y = c_1 \cos(2x) + c_2 \sin(2x) \).
Answer: \( y = c_1 \cos(2x) + c_2 \sin(2x) \).
Example 2: Repeated Roots
Solve \( y'' - 6y' + 9y = 0 \).
- Characteristic: \( r^2 - 6r + 9 = 0 \), \( (r-3)^2 = 0 \), \( r = 3 \) (double).
- Solution: \( y = (c_1 + c_2 x) e^{3x} \).
Answer: \( y = (c_1 + c_2 x) e^{3x} \).
3) Non-Homogeneous Equations 📐
For \( a_n y^{(n)} + \cdots = f(x) \), use a particular solution plus the homogeneous solution.
Definition 28.2: Particular Solution
Use undetermined coefficients or variation of parameters for \( f(x) \) (e.g., polynomials, exponentials).
Example 3: Undetermined Coefficients
Solve \( y'' + y = x \).
- Homogeneous: \( r^2 + 1 = 0 \), \( r = \pm i \), \( y_h = c_1 \cos x + c_2 \sin x \).
- Particular: \( y_p = Ax + B \), \( y_p' = A \), \( y_p'' = 0 \).
- \( 0 + (Ax + B) = x \), \( A = 1 \), \( B = 0 \).
- Solution: \( y = c_1 \cos x + c_2 \sin x + x \).
Answer: \( y = c_1 \cos x + c_2 \sin x + x \).
Example 4: Variation of Parameters
Solve \( y'' + y = \tan x \).
- \( y_h = c_1 \cos x + c_2 \sin x \).
- \( y_p = u_1 \cos x + u_2 \sin x \), \( u_1' \cos x + u_2' \sin x = 0 \), \( -u_1' \sin x + u_2' \cos x = \tan x \).
- Solve: \( u_1' = -\sin x \tan x \), \( u_2' = \sec x \).
- \( u_1 = \sin x \), \( u_2 = \ln|\sec x + \tan x| \).
- \( y_p = \sin x \ln|\sec x + \tan x| \).
- Solution: \( y = c_1 \cos x + c_2 \sin x + \sin x \ln|\sec x + \tan x| \).
Answer: \( y = c_1 \cos x + c_2 \sin x + \sin x \ln|\sec x + \tan x| \).
4) Systems of ODEs 🔍
Systems involve multiple equations, often written as \( \mathbf{y}' = A \mathbf{y} + \mathbf{f}(t) \).
Definition 28.3: Linear System
Solve \( \mathbf{y}' = A \mathbf{y} \) using eigenvalues of \( A \). General solution: \( \mathbf{y} = c_1 \mathbf{v}_1 e^{\lambda_1 t} + c_2 \mathbf{v}_2 e^{\lambda_2 t} \).
Example 5: Two-Variable System
Solve \( \begin{cases} x' = x + y \\ y' = -x + y \end{cases} \).
- Matrix \( A = \begin{pmatrix} 1 & 1 \\ -1 & 1 \end{pmatrix} \).
- Characteristic: \( \det(A - \lambda I) = 0 \), \( (1-\lambda)^2 + 1 = 0 \), \( \lambda = 1 \pm i \).
- Eigenvector for \( \lambda = 1 + i \): \( \mathbf{v} = (1, i) \).
- Solution: \( \mathbf{y} = e^t (c_1 \cos t + c_2 \sin t) (1, 0) + e^t (c_3 \cos t + c_4 \sin t) (0, 1) \).
Answer: \( x = e^t (c_1 \cos t + c_2 \sin t) \), \( y = e^t (c_3 \cos t + c_4 \sin t) \).
Example 6: Non-Homogeneous System
Solve \( \begin{cases} x' = -y + e^t \\ y' = x \end{cases} \).
- Homogeneous: \( \lambda^2 + 1 = 0 \), \( \lambda = \pm i \), \( \mathbf{y}_h = c_1 \cos t (1, 0) + c_2 \sin t (0, 1) \).
- Particular: \( \mathbf{y}_p = (A e^t, B e^t) \), solve \( A = 1 \), \( B = 1 \).
- Solution: \( x = c_1 \cos t + c_2 \sin t + e^t \), \( y = -c_1 \sin t + c_2 \cos t + e^t \).
Answer: \( x = c_1 \cos t + c_2 \sin t + e^t \), \( y = -c_1 \sin t + c_2 \cos t + e^t \).
5) Applications and Phase Planes 🔍
Applications include damped oscillations; phase planes visualize solutions.
Definition 28.4: Phase Plane
Plot \( (x, y) \) vs. \( (x', y') \) to analyze stability.
Example 7: Damped Oscillator
Solve \( y'' + 2y' + y = 0 \).
- Characteristic: \( r^2 + 2r + 1 = 0 \), \( (r+1)^2 = 0 \), \( r = -1 \) (double).
- Solution: \( y = (c_1 + c_2 t) e^{-t} \).
Answer: \( y = (c_1 + c_2 t) e^{-t} \).
Example 8: Phase Plane Analysis
Analyze \( x' = y \), \( y' = -x \).
- Solution: \( x = c_1 \cos t + c_2 \sin t \), \( y = -c_1 \sin t + c_2 \cos t \).
- Phase plane: Circles \( x^2 + y^2 = c^2 \), periodic motion.
Answer: Circular trajectories.
6) Practice Questions 🎯
Fundamental Practice Questions 🌱
Instructions: Solve the differential equations. 📚
\( y'' + 9y = 0 \)
\( y'' - 4y' + 4y = 0 \)
\( y'' + 2y' + 5y = 0 \)
\( y'' + y = \sin x \)
\( y''' - y' = 0 \)
\( \begin{cases} x' = x \\ y' = -y \end{cases} \)
\( y'' + 4y = e^x \)
\( \begin{cases} x' = y \\ y' = -x - y \end{cases} \)
\( y'' - 2y' + y = x^2 \)
\( \begin{cases} x' = 2x + y \\ y' = x + 2y \end{cases} \)
\( y''' + y' = \cos x \)
Challenging Practice Questions 🌟
Instructions: Solve these advanced ODEs or systems. 🧠
Solve \( y''' - 6y'' + 11y' - 6y = 0 \) with initial conditions \( y(0) = 1 \), \( y'(0) = 0 \), \( y''(0) = 0 \).
Find the general solution to \( \begin{cases} x' = x - 2y + e^t \\ y' = 2x - y \end{cases} \).
Determine the phase plane behavior of \( \begin{cases} x' = -y - x^3 \\ y' = x \end{cases} \).
Solve \( y^{(4)} - y = 0 \) and classify the nature of solutions.
Compute the solution to \( \begin{cases} x' = -y + \sin t \\ y' = x + \cos t \end{cases} \) with \( x(0) = 1 \), \( y(0) = 0 \).
7) Summary & Cheat Sheet 📋
7.1) Higher-Order ODEs
Solve using characteristic equation; add particular solution for non-homogeneous.
7.2) Systems of ODEs
Use eigenvalues and eigenvectors; phase planes for behavior.
7.3) Applications
Model oscillations, circuits, and dynamics.
You’ve mastered higher-order ODEs! Next, we’ll explore functional equations. 🎉