Topic 7.2 Notes – Verifying Solutions for Differential Equations

What It Means to Be a Solution to a Differential Equation

A differential equation is any equation that relates a function $y$ and one or more of its derivatives like $\frac{dy}{dx}$ or $y''$.

A function is a solution if, after you:

• Compute the necessary derivative(s), and
• Substitute them into the equation,

the equation is true for all $x$ in the domain.

Think of the differential equation as a rule about how a function must behave. If the function’s derivatives obey that rule, it qualifies.

Quick example:

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

Test $y = 3x^2 + 4$.

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

Substitute into the equation: left side is $6x$, right side is $6x$.
They match, so it works.

That’s the whole idea. No solving. Just checking.

The Verification Process

When you’re asked to verify, it’s always this same structure.

1. Differentiate the proposed function

Use whatever rules are needed:

• Power rule
• Product rule
• Chain rule
• Exponentials and trig
• Possibly second derivatives

This is where most mistakes happen. A small derivative error ruins everything.

2. Substitute into the differential equation

Replace:

• Every $y$ with the original function
• Every $\frac{dy}{dx}$ (or $y''$) with what you computed

If the equation involves both $y$ and $\frac{dy}{dx}$, you must substitute both.

Example:

Verify $y = Ce^{3x}$ satisfies
$$\frac{dy}{dx} = 3y$$

Differentiate:
$$\frac{dy}{dx} = 3Ce^{3x}$$

Substitute into the right side:
$$3y = 3(Ce^{3x}) = 3Ce^{3x}$$

Both sides match. Verified.

3. Simplify completely

Sometimes expressions look different but are algebraically equal.

• Factor if needed
• Rearrange terms
• Combine like terms

Don’t stop early and assume it’s wrong.

On an FRQ, you must clearly show the derivative and substitution steps to earn full credit.

General Solutions and Families of Curves

Most differential equations have infinitely many solutions.

When you solve one, you usually get a general solution that includes a constant $C$.

Example:
$$y = x^4 + C$$

Every different value of $C$ gives a different curve. But they all satisfy the same differential equation.

Each curve is vertically shifted, but their derivatives are identical. That’s why they all satisfy the same equation.

Key ideas:

• When differentiating, treat $C$ as a constant.
• $\frac{d}{dx}(C) = 0$
• If the equation works with $C$ still present, the entire family is verified.

A particular solution happens when a specific value of $C$ is chosen, usually from an initial condition. The verification process is exactly the same.

Common Forms You’ll See

1. Derivative equals a function of x

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

Differentiate the candidate and check if it equals $f(x)$.

2. Derivative equals a function of x and y

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

Differentiate, then substitute both $y$ and $\frac{dy}{dx}$.

Students often forget to plug in $y$ on the right side.

3. Second derivative equations

$$y'' = f(x)$$

Take two derivatives before substituting.

These show up less often but are fair game.

Common Mistakes That Cost Points

• Dropping a chain rule factor, like forgetting the 5 in $e^{5x}$.
• Messing up trig signs. $\frac{d}{dx}(\cos x) = -\sin x$.
• Substituting only the derivative but not the original $y$.
• Not simplifying enough before deciding it doesn’t work.
• Forgetting that constants differentiate to zero in general solutions.

Most verification questions are straightforward. The difficulty is clean execution.