What It Means to Be a Solution
A differential equation is an equation involving an unknown function and one or more of its derivatives. For example, is a differential equation because it relates the function to its derivative.
When a problem gives you a function and asks whether it is a solution to a differential equation, you are not being asked to solve the differential equation from scratch. You are being asked to test the proposed function.
A function is a solution if, after finding the necessary derivatives and substituting them into the differential equation, the equation is true over the relevant interval.
The key idea is simple: differentiate, substitute, simplify, and check.
The Verification Process
There is a reliable four-step method you can use on almost every AP Calculus verification problem.
- Write the proposed solution clearly. Identify the function you are being asked to test.
- Find the derivative or derivatives required by the differential equation. If the equation contains , you need the first derivative. If it contains , you need the second derivative.
- Substitute the function and its derivatives into the differential equation. Replace every occurrence of , , and so on with the appropriate expressions.
- Simplify both sides and determine whether the equation is true. If the two sides are equal, the function is a solution. If they are not equal, it is not a solution.
Worked Examples
Determine whether is a solution to .
Determine whether is a solution to .
Determine whether is a solution to .
Consider . Determine whether is a solution to the differential equation and initial condition .
Checking an Initial Condition
Sometimes the problem gives more than just a differential equation. It may also specify an initial condition, such as .
In that situation, you must perform two separate checks:
- Verify that the function satisfies the differential equation.
- Verify that the function satisfies the initial condition.
A function can satisfy the differential equation but fail the initial condition. If that happens, it is a solution to the differential equation, but not a solution to the particular initial value problem.
This distinction is important because a differential equation can have many solutions, while an initial condition can select one particular member of that family.
Verification Is Not the Same as Solving
There is an important distinction between solving a differential equation and verifying a solution.
When solving, you start with the differential equation and work toward finding an unknown function. When verifying, the function is already given to you. Your job is simply to determine whether it works.
On an AP Calculus problem, this means you usually do not need separation of variables, integration, or any other differential-equation solving technique unless the question specifically asks you to solve the equation.
Using a Given Differential Equation Efficiently
Suppose you are given a function involving a complicated expression and the differential equation only requires its first derivative. Do not calculate derivatives you do not need.
For example, if the equation is , you only need . There is no reason to calculate .
Likewise, if the differential equation contains , you must calculate both the first and second derivatives before substituting.
Common Mistakes
A function satisfying does not automatically satisfy the differential equation. The initial condition is only one part of the verification. You must also substitute the function and its derivatives into the differential equation.
If the differential equation contains , you need the derivative of the proposed function. Do not substitute the original function into every part of the equation.
The two sides may look different before simplification but become identical after algebraic manipulation. Always simplify far enough to make a definite comparison.
A solution must satisfy the differential equation throughout the relevant interval, not merely at one convenient value of . Checking one point is generally not enough to prove that a function is a solution.
Practice Problems
Determine whether is a solution to .
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Determine whether is a solution to .
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Determine whether is a solution to .
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Determine whether is a solution to .
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Determine whether satisfies both the differential equation and the initial condition .
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The AP Calculus Checklist
When you see “verify that the function is a solution”, use this checklist:
- Write the proposed function.
- Calculate every derivative that appears in the differential equation.
- Substitute the function and derivatives into the equation.
- Simplify both sides.
- State explicitly whether the equation is satisfied.
- If an initial condition is given, check it separately.
The final written conclusion matters. Instead of simply ending after an algebraic calculation, write something like “Therefore, the function satisfies the differential equation and is a solution.”
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