Topic 7.2 Notes – Verifying Solutions for Differential Equations

What It Means to Be a Solution to a Differential Equation

A differential equation is an equation that involves a function and its derivative. In AP Calculus AB, it’s usually first order, like

$$\frac{dy}{dx} = f(x, y)$$

A function $y = g(x)$ is a solution if:

• You compute its derivative $g'(x)$,
• Plug both $y$ and $\frac{dy}{dx}$ into the equation,
• And the equation is true for all x in its domain.

That last part matters. It’s not enough for it to work at one value of $x$. It has to work everywhere it’s defined.

So mentally, think:

A solution is just a function that makes the differential equation true.

Verification is a derivative-and-substitute check. Nothing mysterious.

General Solutions and Families of Curves

Most differential equations don’t have just one solution. They have infinitely many.

These are called general solutions, and they usually look like:

$$y = F(x) + C$$

That $C$ represents a whole family of curves.

Here’s a concrete example. Suppose the solutions look like $y = x^2 + C$.

Each curve on the graph has a different value of $C$. They’re all vertical shifts of the same basic parabola, and they all satisfy the same differential equation.

• Different value of $C$,
• Slightly different graph,
• Same differential equation.

If no initial condition is given, you’re dealing with the entire family.

If something like $y(1) = 4$ is given, that picks one specific value of $C$, which gives a particular solution.

On quizzes and FRQs, you’re often asked to verify a solution that includes $C$. Your job is to show it works for any constant.

How to Verify a Solution

When you’re given a differential equation and a proposed function, the process is always the same.

1. Differentiate the Function

Use the correct rules:

• Power rule
• Product rule
• Chain rule
• Exponential/trig derivatives

Most errors happen here. Especially with:

• $e^{kx}$ (don’t forget the $k$)
• Products like $x^2 \cos x$
• Composite trig functions

2. Substitute Into the Equation

Plug in:

• Your derivative for $\frac{dy}{dx}$
• The original function for $y$, if it appears

Do not skip substituting $y$ if the equation includes it.

3. Simplify Completely

Expressions don’t have to look identical at first glance. They must be algebraically equivalent.

For example, rearranged terms are fine:

• $4x + x^2$
• $x^2 + 4x$

If both sides match after simplification, it’s verified.

Example Walkthrough

Suppose:

$$\frac{dy}{dx} = 4x - 3$$

Proposed solution:

$$y = 2x^2 - 3x + 5$$

Step 1: Differentiate

$$\frac{dy}{dx} = 4x - 3$$

Step 2: Compare

The derivative matches the right-hand side exactly.

So the function satisfies the differential equation.

Notice something important. The constant 5 disappeared when we differentiated. That’s why constants show up in general solutions. They don’t affect the derivative.

Verifying a General Solution with $C$

Suppose:

$$\frac{dy}{dx} = 6x$$

Proposed solution:

$$y = 3x^2 + C$$

Differentiate:

$$\frac{dy}{dx} = 6x$$

The $C$ disappears. The derivative matches the equation for any value of $C$.

That means every curve in the family is a solution.

If an initial condition were added, you would solve for $C$. But verification alone does not require finding $C$.

Common Mistakes That Cost Points

• Forgetting the chain rule on $e^{5x}$ or $\sin(3x)$
• Dropping a term in the product rule
• Only plugging in $\frac{dy}{dx}$ but forgetting to plug in $y$
• Checking equality at one x-value instead of algebraically
• Not simplifying fully before comparing

On FRQs, if you don’t show the derivative step clearly, you can lose credit even if your conclusion is correct.