Introduction to Differential Equations and Verification
Define differential equations, verify solutions by differentiation, and understand general vs. particular solutions. Classify ODEs by order. Recognize the types of DEs solvable at the AB level.
A lesson of AP Calculus AB, in Differential Equations, about 10 minutes.
Introduction to Differential Equations
By the end of this lesson, you will be able to identify differential equations, verify solutions, and understand the relationship between general and particular solutions.
What Is a Differential Equation?
A differential equation (DE) is an equation involving a function and one or more of its derivatives. Rather than solving for a number, we solve for a function.
Examples:
- $\frac{dy}{dx} = 2x$ — the simplest type
- $\frac{dy}{dx} = ky$ — exponential growth/decay
- $\frac{d^2y}{dx^2} + 4y = 0$ — second-order DE (beyond AB scope)
A solution to a DE is a function $y = f(x)$ that satisfies the equation.
Verifying a Solution
To verify that $y = f(x)$ is a solution:
- Compute the derivative(s) of $f(x)$
- Substitute into the DE
- Check that both sides are equal
Example: Verify that $y = e^{3x}$ solves $\frac{dy}{dx} = 3y$.
$\frac{dy}{dx} = 3e^{3x}$; $3y = 3e^{3x}$ ✓
Example: Verify that $y = \sin(2x)$ solves $\frac{d^2y}{dx^2} + 4y = 0$.
$y' = 2\cos(2x)$; $y'' = -4\sin(2x)$
$y'' + 4y = -4\sin(2x) + 4\sin(2x) = 0$ ✓
General Solutions vs. Particular Solutions
General solution: the family of all solutions, containing an arbitrary constant $C$.
Particular solution: a specific solution obtained by applying an initial condition (given value of $y$ at a specific $x$).
Example: Solve $\frac{dy}{dx} = 2x$.
Integrate both sides: $y = x^2 + C$ — this is the general solution.
With initial condition $y(1) = 5$: $5 = 1 + C$ → $C = 4$.
Particular solution: $y = x^2 + 4$.
Initial Value Problems
An initial value problem (IVP) is a DE together with an initial condition.
Standard form: $\frac{dy}{dx} = f(x, y)$, $y(x_0) = y_0$
Solution procedure: