Key Concepts in Differential Equations

Why This Matters

Differential equations are the bridge between calculus and the real world—they're how mathematicians and scientists describe anything that changes. When you're tested on this material, you're not just being asked to solve equations; you're being asked to demonstrate that you understand rates of change, modeling, and the relationship between functions and their derivatives. Every physics problem about motion, every population growth model, every circuit analysis comes back to differential equations.

The key to mastering this topic is recognizing patterns: What type of equation am I looking at? What method fits this structure? Don't just memorize solution techniques—understand why each method works and when to apply it. The AP exam loves asking you to identify equation types, select appropriate solving strategies, and interpret solutions in context. If you can classify an equation quickly and match it to the right approach, you're already halfway to the answer.

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Classifying Differential Equations

Before you can solve any differential equation, you need to identify what you're working with. Classification determines method—the wrong approach wastes time and leads nowhere.

Definition of a Differential Equation

• An equation relating a function to its derivatives—this is the foundation of everything in this unit
• Describes rates of change in terms of the variables involved; the derivative represents how fast something changes
• Models real phenomena including motion, growth, decay, and oscillation—expect word problems connecting equations to applications

Order of a Differential Equation

• Determined by the highest derivative present—a second derivative means second-order, regardless of other terms
• First-order equations involve only $\frac{dy}{dx}$, while second-order includes $\frac{d^2y}{dx^2}$
• Higher order = more complex behavior—second-order equations can model oscillations, which first-order cannot

Linear vs. Nonlinear Differential Equations

• Linear equations have the unknown function and its derivatives appearing only to the first power, with no products between them
• Nonlinear equations include terms like $y^2$, $y \cdot \frac{dy}{dx}$, or $\sin(y)$—these break the superposition principle
• Linear equations have predictable solution methods; nonlinear equations often require numerical approaches or special techniques

Ordinary vs. Partial Differential Equations

• ODEs involve functions of one variable—like $\frac{dy}{dx}$ where $y$ depends only on $x$
• PDEs involve multiple independent variables and partial derivatives like $\frac{\partial u}{\partial t}$ and $\frac{\partial u}{\partial x}$
• AP Calculus focuses on ODEs—PDEs appear in multivariable calculus and physics courses

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Problem Types and Conditions

Knowing what conditions you're given determines how you approach the solution. Initial conditions pin down one specific solution from infinitely many possibilities.

Initial Value Problems

• Specifies the function's value at a starting point—typically given as $y(x_0) = y_0$
• Guarantees a unique solution under the conditions of the Existence and Uniqueness Theorem
• Most common problem type on AP exams—you'll solve the general equation, then apply the initial condition to find constants

Boundary Value Problems

• Specifies conditions at two or more points—such as $y(0) = a$ and $y(L) = b$
• Common in physical applications like heat distribution along a rod or vibrations of a string
• Solutions may not exist or may not be unique—this is a key distinction from initial value problems

General and Particular Solutions

• General solution contains arbitrary constants—represents the entire family of solutions to the equation
• Particular solution has specific constant values determined by initial or boundary conditions
• The number of arbitrary constants equals the order of the differential equation; a second-order equation has two constants

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First-Order Solution Methods

First-order equations are your bread and butter. Master these techniques—they appear constantly on exams and form the foundation for more advanced methods.

Separation of Variables

• Rearrange to isolate variables—get all $y$ terms and $dy$ on one side, all $x$ terms and $dx$ on the other
• Works when the equation can be written as $\frac{dy}{dx} = f(x)g(y)$; not all equations are separable
• Integrate both sides independently—don't forget the constant of integration and absolute values in logarithms

First-Order Linear Differential Equations

• Standard form is $\frac{dy}{dx} + P(x)y = Q(x)$—rearrange any first-order linear equation into this form first
• Solved systematically using integrating factors—this method always works for linear equations
• Recognize the pattern—if you see $y$ and $\frac{dy}{dx}$ both to the first power with no products, it's linear

Integrating Factors

• Multiply the entire equation by $\mu(x) = e^{\int P(x)\,dx}$ to make the left side a perfect derivative
• Transforms the equation into $\frac{d}{dx}[\mu(x)y] = \mu(x)Q(x)$, which integrates directly
• The key insight: the integrating factor turns the left side into the derivative of a product; this is reverse product rule

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Second-Order Linear Equations

Second-order equations model systems with acceleration, oscillation, or feedback. The characteristic equation is your primary tool here.

Second-Order Linear Differential Equations

• General form is $a\frac{d^2y}{dx^2} + b\frac{dy}{dx} + cy = g(x)$ where $a$, $b$, $c$ may be constants or functions
• Constant coefficient cases are most common on AP exams and have systematic solution methods
• Model oscillating systems—springs, circuits, and waves all produce second-order equations

Homogeneous and Non-Homogeneous Equations

• Homogeneous means $g(x) = 0$—the equation equals zero, with no "forcing function"
• Non-homogeneous includes a forcing term $g(x) \neq 0$ that drives the system
• Total solution = homogeneous solution + particular solution; you must find both parts separately

Characteristic Equation

• Substitute $y = e^{rx}$ into the homogeneous equation to get a polynomial in $r$
• For $ay'' + by' + cy = 0$, the characteristic equation is $ar^2 + br + c = 0$
• Root types determine solution form: real distinct → $C_1e^{r_1x} + C_2e^{r_2x}$; repeated → $(C_1 + C_2x)e^{rx}$; complex → $e^{\alpha x}(C_1\cos\beta x + C_2\sin\beta x)$

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Finding Particular Solutions

Once you've solved the homogeneous equation, you need techniques for the particular solution of non-homogeneous equations. Choose your method based on what $g(x)$ looks like.

Method of Undetermined Coefficients

• Guess a solution form based on $g(x)$—polynomials suggest polynomial guesses, exponentials suggest exponential guesses
• Substitute your guess into the equation and solve for the unknown coefficients
• Works best for $g(x)$ that is a polynomial, exponential, sine, cosine, or combination; fails for functions like $\tan(x)$ or $\ln(x)$

Variation of Parameters

• Uses the homogeneous solutions $y_1$ and $y_2$ as building blocks for the particular solution
• Formula involves integrals: $y_p = -y_1\int\frac{y_2 g(x)}{W}\,dx + y_2\int\frac{y_1 g(x)}{W}\,dx$ where $W$ is the Wronskian
• More general than undetermined coefficients—works for any continuous $g(x)$, but computations are often messier

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Quick Reference Table

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Self-Check Questions

1. Given the equation $y'' + 4y' + 4y = e^{-2x}$, what type of roots does the characteristic equation have, and how does this affect your guess for the particular solution using undetermined coefficients?

2. Which two solution methods—separation of variables and integrating factors—both work for the equation $\frac{dy}{dx} = \frac{x}{y}$? Why might you prefer one over the other?

3. Compare and contrast initial value problems and boundary value problems: What guarantees uniqueness in one case but not the other?

4. If a second-order linear equation has characteristic roots $r = 3 \pm 2i$, write the general form of the homogeneous solution and explain what physical behavior this represents.

5. An FRQ presents the equation $\frac{dy}{dx} + 2xy = x$ with $y(0) = 1$. Outline your solution strategy: What method would you use, what is the integrating factor, and how would you apply the initial condition?