AP® Calculus AB review sheet from Aim for Five (aimforfive.com/calc-ab/units/7/7-2)
Unit 7 · Topic 7.2
7.2 Verifying Solutions for Differential Equations
A solution to a differential equation is a function that makes the equation true for every x. You check a possible solution by differentiating it, substituting into the equation and seeing whether both sides match.
Key terms
- solution of a differential equation
- verify
- substitution
- general solution
- family of functions
Solutions are functions
When you solve x + 3 = 5, you get a number. When you solve a differential equation, you get a function. The function y = f(x) is a solution if, after you substitute f and its derivatives into the equation, the two sides are equal for every x in an interval.
Usually there are infinitely many solutions. For dy/dx = 2x, any function y = x² + C works, for any constant C. That whole collection is called a family of solutions (or the general solution). Picking one, by requiring the curve to pass through a certain point, gives a particular solution (7.7).
How to verify a solution
- Compute every derivative the equation uses (y′, and y″ if it appears).
- Substitute the function and its derivatives into the left side and the right side separately.
- Simplify each side. If they're identical for all x in the interval, the function is a solution. If they differ, even for one x in the interval, it isn't.
- If an initial condition is given, also check that the function passes through that point.
Second-order equations
Some equations involve y″. The process is the same: compute y′ and y″, substitute and compare. For instance, y = sin(2x) solves y″ + 4y = 0 because y″ = −4 sin(2x), and −4 sin(2x) + 4 sin(2x) = 0.
A common question type asks you to find a constant that makes a function a solution. Substitute the function with the unknown constant, and solve the resulting equation for the constant.
Implicit solutions
Sometimes the proposed solution is an equation in x and y rather than y = f(x). Check it with implicit differentiation (3.2). For example, to check that x² + y² = C solves dy/dx = −x/y, differentiate both sides: 2x + 2y·(dy/dx) = 0, so dy/dx = −x/y. That's exactly the equation, so every circle centered at the origin is a solution curve (for y ≠ 0).
Notice the constant C vanished when you differentiated. That's why one differential equation can have a whole family of solutions: the derivative can't see which constant you picked.
Why this matters
Verification is your safety net. After you solve a separable equation (7.6, 7.7), you can check your answer by plugging it back in. It also helps on multiple-choice questions: rather than solving from scratch, you can test each answer choice in the equation.
Worked examples
Try each one yourself first, then open the solution.
- Example 1
Verify a first-order solution
Show that y = 3e²ˣ − 1 is a solution of dy/dx = 2y + 2.
Show the solutionHide the solution
- Step 1: Left side: dy/dx = 6e²ˣ.
- Step 2: Right side: 2y + 2 = 2(3e²ˣ − 1) + 2 = 6e²ˣ − 2 + 2 = 6e²ˣ.
- Step 3: Both sides equal 6e²ˣ for every x, so the function is a solution.
Answer: Yes: both sides simplify to 6e²ˣ.
- Example 2
Find the constant
For which values of k is y = eᵏˣ a solution of y″ − y′ − 6y = 0?
Show the solutionHide the solution
- Step 1: y′ = keᵏˣ and y″ = k²eᵏˣ.
- Step 2: Substitute: k²eᵏˣ − keᵏˣ − 6eᵏˣ = eᵏˣ(k² − k − 6).
- Step 3: eᵏˣ is never 0, so you need k² − k − 6 = 0, which factors as (k − 3)(k + 2) = 0.
Answer: k = 3 or k = −2
- Example 3
Trap: it must work for every x
Is y = eˣ + 1 a solution of dy/dx = y?
Show the solutionHide the solution
- Step 1: Left side: dy/dx = eˣ.
- Step 2: Right side: y = eˣ + 1.
- Step 3: eˣ ≠ eˣ + 1 for any x, so it isn't a solution.
- Step 4: Students sometimes say “close enough” or check only that the shapes look alike. The two sides must match exactly.
Answer: No. dy/dx = eˣ, which never equals eˣ + 1.
Common mistakes
- Checking only at one point. A solution must satisfy the equation everywhere on its interval.
- Substituting the derivative in for y, or y in for the derivative, by accident. Compute each side separately.
- Forgetting the chain rule when differentiating something like e²ˣ or sin(2x).
- Treating a solution as a number, like y = 4, when the question asks for a function.
On the exam
- Multiple-choice questions often ask which function satisfies a given differential equation. Differentiate each choice and test it.
- Some questions ask which differential equation a given function satisfies. Differentiate the function and rewrite the result in terms of y.
Connected topics
Videos
Check yourself
4 questions on 7.2 Verifying Solutions for Differential Equations. Pick an answer to see if you got it, and why.
Which of the following functions is a solution to the differential equation dy/dx = 2y/x for x > 0?
Which of the following functions are solutions to the differential equation dy/dx = y − x? I. y = x + 1 II. y = eˣ + x + 1 III. y = eˣ − x − 1
For which values of the constant k is y = e^(kx) a solution to the differential equation y″ − 3y′ − 10y = 0?
The function y = ax² + bx, where a and b are constants, is a solution to the differential equation x·(dy/dx) − y = x² for x ≠ 0. Which of the following must be true?
0 of 4 answered