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Verifying Solutions for
Differential Equations

Learn how to substitute a proposed function into a differential equation and confirm it satisfies the equation: the essential first skill in differential equations.

AP Calculus AB

What Is a Differential Equation?

A differential equation is an equation that involves an unknown function and one or more of its derivatives. Instead of solving for a number, you solve for an entire function.

Example
\(\dfrac{dy}{dx} = 2x\) asks: "Which function \(y\) has a derivative equal to \(2x\)?"
Solution
\(y = x^2 + C\) for any constant \(C\). Every such function satisfies the equation.

What Does "Verify" Mean?

To verify a solution means to show that when you substitute the proposed function (and its derivatives) into the differential equation, both sides become identically equal.

Proposed
\(y = f(x)\)
 Differentiate
Find \(y'\)
 Substitute
LHS = RHS ?

Why Verification Matters

Many differential equations have families of solutions. Verification is the tool that lets you confirm a particular candidate belongs to that family: or reject it if it does not.

On the AP Exam
Free-response problems often give a function and ask you to show it satisfies a given DE. Partial credit depends on clear substitution steps.
In Applications
Models in physics and biology produce DEs. Verifying a proposed model ensures your predicted behavior matches the governing equation.

The Verification Principle

Differential Equation
Replace \(y\) with \(f(x)\), replace \(y'\) with \(f'(x)\)
 Simplify both sides
LHS = RHS ✓

This must hold for all \(x\) in the domain, not just specific points. A single counterexample means the function is not a solution.

General vs. Particular Solutions

General Solution
Contains an arbitrary constant \(C\). Represents the entire family of curves. Example: \(y = Ce^{x}\) for \(\dfrac{dy}{dx} = y\).
Particular Solution
A specific member obtained by applying an initial condition. Example: if \(y(0) = 3\), then \(y = 3e^{x}\).

Solving vs. Verifying: Key Distinction

Solving Verifying
Starting point Only the DE is given DE and a proposed \(y\) are given
Goal Find the unknown function \(y\) Show the given \(y\) satisfies the DE
Techniques Separation, integrating factors, etc. Differentiation + substitution
AP Calc AB focus Separation of variables only Frequently tested on FRQs

How to Verify: Step by Step

1
Identify the DE and the proposed solution. Write down the differential equation and the candidate function \(y = f(x)\).
2
Compute all needed derivatives. Find \(y'\), and \(y''\) if the DE involves second derivatives.
3
Substitute into the DE. Replace every occurrence of \(y\), \(y'\), \(y''\) with the expressions you computed.
4
Simplify both sides. Use algebra, trig identities, or exponential rules to reduce.
5
Confirm the identity. If LHS = RHS for all \(x\) in the domain, the function is a verified solution.

Example 1: Basic Verification

1
Find \(y'\): \(y' = \dfrac{d}{dx}(x^2 + 3x) = 2x + 3\)
2
Substitute into the DE: LHS = \(y' = 2x + 3\). RHS = \(2x + 3\).
3
Compare: \(2x + 3 = 2x + 3\) ✓ True for all \(x\).

Example 2: Exponential Function

1
Find \(y'\): \(y' = \dfrac{d}{dx}(Ce^{3x}) = 3Ce^{3x}\)
2
Compute RHS: \(3y = 3 \cdot Ce^{3x} = 3Ce^{3x}\)
3
Compare: LHS = \(3Ce^{3x}\), RHS = \(3Ce^{3x}\). ✓ Identical.

A cell culture where more cells create more growth events

A researcher looks through a microscope beside a closed culture vessel,sample containers and an illustrative cell inset.In the model y = 100e^(0.4t), the relative growth rate stays at 0.4 while total growth y′ = 0.4y increases with the population.

A cell culture where more cells create more growth events

In the model y = 100e^(0.4t), the relative growth rate stays at 0.4 while total growth y′ = 0.4y increases with the population.In the model y = 100e^(0.4t), the relative growth rate stays at 0.4 while total growth y′ = 0.4y increases with the population.

Relative rate versus total rate

In the model y = 100e^(0.4t), the relative growth rate stays at 0.4 while total growth y′ = 0.4y increases with the population.In the model y = 100e^(0.4t), the relative growth rate stays at 0.4 while total growth y′ = 0.4y increases with the population.

Example 3: Second-Order DE

1
Find \(y'\): \(y' = 2\cos(2x)\)
2
Find \(y''\): \(y'' = -4\sin(2x)\)
3
Substitute: \(y'' + 4y = -4\sin(2x) + 4\sin(2x) = 0\)
4
Result: \(0 = 0\) ✓ True for all \(x\).

Visualizing Solution Curves

Consider the DE \(\dfrac{dy}{dx} = y\). Its general solution is \(y = Ce^x\). Below, each curve corresponds to a different value of \(C\).

x y -2 -1 1 2 C = 2 C = 1 C = 0.5 C = -1

When Verification Fails

1
Find \(y'\): \(y' = 2e^{2x}\)
2
Compute RHS: \(3y = 3e^{2x}\)
3
Compare: \(2e^{2x} \neq 3e^{2x}\). The identity does not hold.

Verifying with Initial Conditions

Sometimes you must verify both that the function satisfies the DE and that it passes through a given point \((x_0, y_0)\).

1
Check the DE: \(y' = 2e^{x}\) and \(y = 2e^{x}\), so \(y' = y\). ✓
2
Check the IC: \(y(0) = 2e^{0} = 2 \cdot 1 = 2\). ✓
3
Conclusion: Both conditions satisfied. \(y = 2e^{x}\) is the solution to the IVP.

Common Mistakes to Avoid

✕ Forgetting the Chain Rule
When differentiating \(e^{3x}\), writing \(e^{3x}\) instead of \(3e^{3x}\). Always apply the chain rule to inner functions.
✕ Checking Only One Point
Plugging in \(x = 0\) and seeing it works does not verify for all \(x\). You must show the identity algebraically.
✕ Dropping the Constant \(C\)
When verifying a general solution, keep \(C\) throughout. It should cancel naturally if the solution is correct.

Summary: Verification Checklist

Step Action Watch For
1 Write down DE and proposed \(y\) Note the order of the DE
2 Differentiate \(y\) Chain rule, product rule
3 Substitute into DE Replace every \(y, y', y''\)
4 Simplify both sides Factor, combine like terms
5 Confirm LHS = RHS Must hold for all \(x\)
6 Check IC (if given) Plug in \(x_0\), verify \(y_0\)

Question 1 of 5

Which of the following is a solution to the differential equation \(\dfrac{dy}{dx} = 2y\)?

Question 2 of 5

To verify that \(y = \cos(x)\) is a solution of \(y'' + y = 0\), what is the first step?

Question 3 of 5

A student claims that \(y = x^3\) satisfies \(\dfrac{dy}{dx} = 3x\). Is this correct?

Question 4 of 5

Given the IVP: \(\dfrac{dy}{dx} = -y\), \(y(0) = 4\), which function is the particular solution?

Question 5 of 5

When verifying a general solution \(y = Ce^{kx}\) in a first-order DE, what should happen to the constant \(C\)?