Verifying Solutions for Differential Equations
Differentiate and substitute throughout the domain, checking an initial condition separately.
College Board Topic 7.2 · CED p. 133 (PDF p. 138) · CED effective Fall 2020; project course outline 2026–27.
To verify a proposed solution, differentiate it and substitute into the equation. Both sides must agree for every point in the stated domain. Then check the initial condition separately. Matching at one point proves only a single numerical match, not that a function solves the differential equation.
There may be infinitely many solutions to a differential equation. For y′=2y, the family y=Ce^(2x) works for every constant C. The initial condition y(0)=3 selects C=3 within that family. Do not say that every differential equation always has infinitely many solutions, or that every possible initial-value problem has a unique solution: existence and uniqueness depend on the equation’s regularity. In the standard smooth models here, verify the particular solution and its permitted interval.
Let's work through it
- Test y=3e^(2x) in y′=2y: y′=6e^(2x). Reason: Differentiate the proposed function.
- The right side is 2y=6e^(2x), agreeing for all x. Reason: The equality verifies the differential equation.
- Also y(0)=3, so it satisfies that initial condition. Reason: The initial condition is checked independently.
Try it from memory
Pause & retrieve
Try it before you peek.
Write your answer and the reason from memory. Then compare with the worked answer. Your self-assessment is a study decision, not an automatically graded score.
Question 1
If two sides agree only at x=0, has a solution been verified?
See every recall question with its answer
These references preserve each prompt and its answer for direct evidence links. Try the interactive questions above before opening the worked reasoning.
Retrieval 1
If two sides agree only at x=0, has a solution been verified?
Worked answer and reason
No.
The differential equation must hold throughout the domain.
Retrieval 2
What does the constant C in y=Ce^(2x) represent?
Worked answer and reason
Different solution curves with different initial values.
Each C gives a function satisfying the same differential equation.
Catch the tempting mistake
- A graph that looks similar is enough to verify a solution.
- Differentiate and substitute. Graphs help interpret, but an identity verifies the equation.
Calculator workspace and limits
For calculus, set the angle selector to Radians. This scientific calculator evaluates arithmetic and functions. It does not graph arbitrary functions, find numerical roots, evaluate numerical derivatives, or compute definite integrals. Those tasks need an approved graphing tool; the May 2027 exam permits approved handheld graphing calculators and Bluebook’s built-in Desmos on calculator-required parts. A graph or decimal estimate helps you check a conclusion; your setup, hypotheses, units and justification still do the explaining.
Scientific calculator
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Put it to work
- Unit 7 problem 4 · No calculator · analytic
- Unit 7 problem 5 · No calculator · analytic
- Unit 7 problem 6 · No calculator · analytic
For the teacher
- Use one correct and one plausible wrong candidate. Emphasize the Fall 2026 clarification: a differential equation may have infinitely many solutions.
Glossary and deeper reading · 4 refreshers
These editorial refreshers supplement the reading. Definitions and examples are added guidance; original source wording and recorded answers remain in their original context.
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Domain
The domain of a function is the set of inputs for which the function is defined. An expression and its context can both restrict the allowed inputs. For example, the real function f(x)=1/(x−2) has domain all real numbers except 2. For a model of elapsed time, a sensible domain may also require t≥0.
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Initial condition
An initial condition specifies a solution’s value at a particular input. It can determine the constant in a family of solutions to a differential equation. For example, y′=2x with y(0)=3 selects y=x²+3. The solution must satisfy both the equation and the stated condition on a suitable interval.
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Differential equation
A differential equation relates an unknown function to one or more of its derivatives. A solution is a function that makes the equation true throughout an appropriate interval. For example, y′=2x has solutions y=x²+C. An initial condition can select a particular member of a family of solutions.
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Particular solution
A particular solution is one specific function satisfying a differential equation. Initial conditions often select it from a general family. For example, y=x²+3 is the solution of y′=2x with y(0)=3. Verify the selected function in both the differential equation and the initial condition.