AP Syllabus focus:
‘To verify that a function is a solution of a differential equation, students compute the derivative and substitute both the function and its derivative into the equation to check that the relationship is satisfied.’
Verifying solutions to differential equations ensures a proposed function truly models a described relationship. This process relies on differentiation, substitution, and confirming that both sides of the equation match correctly.
Checking Solutions Using Derivatives
To check whether a given function satisfies a differential equation, students rely on the fundamental connection between a function and its rate of change. A differential equation is an equation that includes an unknown function and one or more of its derivatives. Before verifying any proposed solution, it is essential to understand how the derivative behaves in relation to the original function and how the differential equation expresses a relationship that the solution must follow.
Differential Equation: An equation that relates an unknown function to one or more of its derivatives.
Verifying a solution involves determining whether the function and its derivative satisfy the stated relationship at all points where the function is defined. This process is central in AP Calculus AB because it reinforces both symbolic fluency with derivatives and conceptual understanding of rate-based relationships.
Purpose of Verifying Solutions
The act of checking solutions ensures that a proposed function truly models the situation expressed by the differential equation. A function that merely looks plausible might fail when substituted into the equation. Students learn that a valid solution must fulfill the differential equation identically, meaning the equality must hold for all relevant values of the independent variable.
The Verification Process
The verification procedure follows a consistent, structured pattern.
Several solution curves for the differential equation are shown, each representing a distinct function satisfying the same relationship. The curves illustrate how multiple functions can all fulfill the differential equation at every point in their domains. The presence of several curves goes slightly beyond the basic requirement of verifying a single proposed solution. Source.
Students should be able to articulate and perform each step clearly.
Key steps in verifying a solution:
Identify the given differential equation and note which derivative it involves.
Compute the necessary derivative of the proposed function using appropriate differentiation rules.
Substitute the proposed function into the original equation wherever the dependent variable appears.
Substitute the computed derivative wherever the derivative expression appears.
Determine whether both sides of the differential equation match for all valid input values.
Students must also understand that verifying involves substitution into the entire differential equation, not into isolated parts. This ensures that the function aligns with the specific rate-of-change relationship described by the problem.
= Derivative of with respect to , representing instantaneous rate of change
= Expression that the derivative must equal, determined by the differential equation
After introducing this basic form, students can connect how verifying requires substituting both components — the derivative and the original function — into their respective positions. Recognizing how the differential equation prescribes the slope of any valid solution is crucial for interpreting the results of the substitution step.
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Interpreting the Outcome of Verification
Once substitution is complete, students evaluate whether the equality holds:
If the expression obtained from substituting the derivative equals the expression obtained from substituting the function, the proposed function is a solution.
If the expressions do not match, even at a single point in the function’s domain, the proposed function is not a solution.
This reinforces an important distinction: verifying a solution is not about solving the equation but confirming whether a given candidate already satisfies it.
Solution to a Differential Equation: A function whose derivative and original expression satisfy the differential equation for all values in its domain.
For AP Calculus AB, the emphasis is on clear symbolic substitution and logical justification. The student must demonstrate that the derivative calculation is correct and that the substitution was performed precisely as the equation dictates.
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What Students Should Demonstrate
When checking solutions, students are expected to:
Show accurate differentiation using rules appropriate to the given function (power, product, quotient, exponential, or trigonometric rules).
Substitute carefully and consistently, keeping dependent and independent variables distinct.
Present a clear statement indicating whether the function satisfies the differential equation based on the resulting equality.
Maintain correct notation, especially with derivative symbols such as , ensuring alignment with AP standards.
Students should recognize that verifying solutions is conceptually tied to modeling real-world behavior. Because differential equations describe how a quantity changes, verifying a solution tests whether a proposed function genuinely represents that changing behavior. Understanding this connection supports later work with slope fields, separable equations, and applied modeling.
The diagram shows how a differential equation combines an unknown function and its derivative in one relationship. The annotations identify , , and the integrating factor , reflecting typical expressions involved when verifying or manipulating differential equations. The additional solving steps extend beyond the basic requirement of verifying a proposed solution. Source.
FAQ
A quick way is to compare the general form of the function with the structure of the differential equation.
If the equation involves polynomial expressions, exponential terms, or trigonometric forms, check whether the proposed function has a compatible structure.
This does not prove anything, but it helps you anticipate whether the substitution is likely to produce matching expressions.
You can also look at whether the function’s derivative will create terms that resemble the right-hand side. This makes the verification more intuitive before you begin working formally.
A valid solution must satisfy the differential equation identically, meaning it must hold for all x-values in its domain.
Checking only selected points may hide mismatches that occur elsewhere.
Even if the equality holds at multiple points, that does not guarantee the function follows the required rate-of-change relationship throughout the interval.
The differential equation defines an entire pattern of behaviour, not isolated instances, so full algebraic verification is essential.
Students often confuse the function with its derivative, placing expressions in the wrong part of the equation.
Other frequent errors include:
Forgetting to substitute the function y where required.
Only substituting the derivative but not the function values.
Simplifying incorrectly after substitution, which may produce a false mismatch.
Careful organisation of the substitution step helps prevent these issues.
Yes. Even if the algebraic substitution appears to work, the function might not qualify if its domain does not align with the domain required by the differential equation.
If the differential equation includes expressions that restrict possible x-values (such as denominators or square roots), the proposed function must be defined wherever the equation needs it to be evaluated.
A mismatch in domain prevents the function from being considered a valid solution.
Verification ensures that any proposed solution—whether guessed, derived from context, or produced by technology—truly satisfies the equation’s rate-of-change rule.
This is especially useful when:
Modelling situations where a candidate function seems plausible.
Checking the correctness of solutions obtained through integration or algebraic manipulation.
Confirming that a function is part of the family of solutions without solving the entire differential equation.
Verification is a universal check independent of method, reinforcing mathematical accuracy.
Practice Questions
Question 1 (1–3 marks)
A function is defined by f(x) = 3x² – 4x. Consider the differential equation dy/dx = 6x – 4.
Determine whether f(x) is a solution to the differential equation. Show working.
Mark scheme:
• 1 mark for correctly differentiating f(x) to obtain f ’(x) = 6x – 4.
• 1 mark for substituting f ’(x) into the differential equation.
• 1 mark for concluding that f(x) is a solution because its derivative matches the right-hand side of the differential equation for all x.
Question 2 (4–6 marks)
Let g(x) = e^(2x) – 5. Consider the differential equation dy/dx = 2y + 10.
(a) Verify that g(x) is a solution to the differential equation.
(b) Explain why the verification process confirms that the relationship holds for all x in the domain of g.
Mark scheme:
• 1 mark for correctly differentiating g(x) to obtain g ’(x) = 2e^(2x).
• 1 mark for substituting g(x) into the expression 2y + 10 to obtain 2(e^(2x) – 5) + 10.
• 1 mark for simplifying the expression to show that it equals 2e^(2x).
• 1 mark for concluding that g ’(x) equals the right-hand side of the differential equation, verifying that g(x) is a solution.
• 1–2 marks for a clear explanation in part (b), noting that the equality holds for all x in the domain and therefore g(x) satisfies the differential equation identically.
