Nonhomogeneous Linear Second-Order Differential Equations

This study guide covers the techniques for solving second-order linear differential equations where the right-hand side is non-zero. These equations are fundamental in modeling physical systems with external forces, such as forced harmonic motion or RLC circuits.

Learning Objectives

After studying this material, you should be able to:

• Distinguish between homogeneous and nonhomogeneous linear differential equations.
• Construct the general solution to a nonhomogeneous equation using the principle of superposition.
• Apply the Method of Undetermined Coefficients for polynomial, exponential, and trigonometric forcing functions.
• Utilize the Method of Variation of Parameters for more complex nonhomogeneous terms.
• Solve initial-value problems (IVPs) for nonhomogeneous systems.

Key Terms & Glossary

• Nonhomogeneous Term ($g(x)$): The non-zero function on the right side of the equation ay'' + by' + cy = g(x). Also called the "forcing function."
• Complementary Solution ($y_c$): The general solution to the associated homogeneous equation ($g(x) = 0$). It represents the natural response of the system.
• Particular Solution ($y_p$): Any single solution that satisfies the full nonhomogeneous equation.
• Superposition Principle: The theorem stating that the general solution is the sum of the complementary and particular solutions: $y(x) = y_c(x) + y_p(x)$.
• Wronskian ($W$): A determinant used in the Method of Variation of Parameters to ensure the linear independence of solutions.

The "Big Idea"

In physics and engineering, a nonhomogeneous equation represents a system acted upon by an external force. The complementary solution ($y_c$) tells us how the system behaves on its own (the "transient" behavior), while the particular solution ($y_p$) tells us how the system responds specifically to the external driver (the "steady-state" behavior). To find the complete story of the system's motion, you must combine both.

Formula / Concept Box

Hierarchical Outline

1. Introduction to Nonhomogeneous Equations
   • Definition: $L[y] = g(x)$ where $g(x) \neq 0$.
   • The Complementary Equation: $L[y] = 0$.
2. Method of Undetermined Coefficients
   • Applicability: Works when $g(x)$ is a polynomial, $e^{rx}$, $\sin(kx)$, or $\cos(kx)$.
   • The Guess: Assume $y_p$ takes the same form as $g(x)$.
   • The Overlap Rule: If $g(x)$ is part of $y_c$, multiply the guess by $x^n$ until no terms overlap.
3. Method of Variation of Parameters
   • Applicability: More general; works for any continuous $g(x)$.
   • Formula: $y_p = u_1 y_1 + u_2 y_2$.
4. Applications
   • Forced Vibrations: $m x'' + c x' + k x = F_0 \cos(\omega t)$.
   • RLC Circuits: $L Q'' + R Q' + \frac{1}{C} Q = E(t)$.

Visual Anchors

Decision Flow: Choosing a Method

Physical Intuition: Forced Oscillation

Definition-Example Pairs

• Term: Undetermined Coefficients Guess

• Definition: Selecting a template for $y_p$ based on the derivatives of $g(x)$.

• Example: If $g(x) = 3x^2$, the guess is $y_p = Ax^2 + Bx + C$. If $g(x) = 5e^{2x}$, the guess is $y_p = Ae^{2x}$.

• Term: Variation of Parameters

• Definition: A method that replaces constants $c_1, c_2$ with functions $u_1(x), u_2(x)$.

• Example: Used for $g(x) = \sec(x)$ or $\tan(x)$, which cannot be modeled by simple coefficients.

Worked Examples

Example 1: Undetermined Coefficients

Problem: Solve $y'' - 3y' + 2y = 4e^{3x}$.

1. Find $y_c$: Solve $r^2 - 3r + 2 = 0 \implies (r-1)(r-2)=0$. So $y_c = c_1 e^x + c_2 e^{2x}$.
2. Guess $y_p$: $g(x) = 4e^{3x}$. There is no overlap with $y_c$, so let $y_p = Ae^{3x}$.
3. Differentiate: $y_p' = 3Ae^{3x}$, $y_p'' = 9Ae^{3x}$.
4. Substitute: $(9Ae^{3x}) - 3(3Ae^{3x}) + 2(Ae^{3x}) = 4e^{3x}$.
5. Simplify: $9A - 9A + 2A = 4 \implies 2A = 4 \implies A = 2$.
6. Final Solution: $y = c_1 e^x + c_2 e^{2x} + 2e^{3x}$.

Example 2: Variation of Parameters

Problem: Solve $y'' + y = \tan(x)$.

1. Find $y_c$: $r^2 + 1 = 0 \implies r = \pm i$. $y_1 = \cos(x), y_2 = \sin(x)$.
2. Wronskian: $W = (\cos x)(\cos x) - (\sin x)(-\sin x) = \cos^2 x + \sin^2 x = 1$.
3. Find $u_1, u_2$:
   • $u_1 = -\int \frac{\sin x \tan x}{1} dx = -\int \frac{\sin^2 x}{\cos x} dx = -\int (\sec x - \cos x) dx = -\ln|\sec x + \tan x| + \sin x$.
   • $u_2 = \int \frac{\cos x \tan x}{1} dx = \int \sin x dx = -\cos x$.
4. $y_p$ Calculation: $y_p = u_1 y_1 + u_2 y_2 = [-\ln|\sec x + \tan x| + \sin x]\cos x + [-\cos x]\sin x = -\cos x \ln|\sec x + \tan x|$.

Checkpoint Questions

1. If $y_c = c_1 e^{2x} + c_2 x e^{2x}$ and $g(x) = e^{2x}$, what should your guess for $y_p$ be? (Hint: check for overlap).
2. Why is the Method of Undetermined Coefficients generally preferred over Variation of Parameters when $g(x)$ is a polynomial?
3. True or False: The general solution to $y'' + 4y = \sin(2x)$ will contain a term multiplied by $x$ because of resonance.

[!TIP] When performing Variation of Parameters, always ensure your differential equation is in standard form ($y''$ coefficient is 1) before identifying $g(x)$.

▶Click to expand answers

1. $y_p = Ax^2 e^{2x}$ (since $e^{2x}$ and $xe^{2x}$ are already in $y_c$).
2. It avoids complex integration and involves only algebraic manipulation.
3. True. Since $\sin(2x)$ is a solution to the homogeneous equation $y'' + 4y = 0$, you must multiply the guess by $x$.