How can you check a proposed solution without solving the equation?
You will be able to: Verify the differential equation throughout a domain and check the initial value separately.
How can you check a proposed solution without solving the equation?
A friend proposes y=3e^x for the rule y′=y and starting value y(0)=2. The function follows the growth rule, but does it start in the right place?
A useful starting point: How is a constant rate different from proportional change? →
Words and symbols before equations
- Candidate solution
- A function proposed to satisfy the equation.
- Residual
- Left side minus right side after substitution.
- Solution family
- Several functions satisfying the same equation.
- Initial-value problem
- A differential equation together with starting data.
What this picture assumes
Original model; rounded readouts and finite sampled graphs. Equation y′=y with initial condition y(0)=2. Axes dimensionless. A sampled residual is illustrative; the algebra checks the identity.
Read the picture in three steps
- Read the axes and labels first. Identify what each symbol and line represents. Read the units and fixed conditions before comparing quantities.
- Equation holds algebraically; initial value is 2, matching required 2. Navy y; orange dashed y′.
- Check what the picture assumes below. Use the Explore task to predict one change before moving a control.
Connect the picture to the mathematics
Differentiate the candidate first, then substitute the candidate and its derivative into the original equation. Equality must hold throughout the stated interval, not just at one point.
For y=Ce^x, y′=Ce^x=y for every real x and any constant C. Thus this equation has infinitely many solutions; this is possible, not a claim about every equation.
The initial condition is a separate test. y=3e^x gives y(0)=3, so it fails the given initial value 2. The particular solution is 2e^x.
A sampled residual graph can reveal a mismatch but cannot prove an identity everywhere. Algebra supplies the verification. Candidates must also be defined and differentiable on the proposed solution interval.
A worked example, step by step
Verify y=x²+2x+2 for y′=y−x² with y(0)=2.
- Differentiate to obtain y′=2x+2.
- Substitute on the right: y−x²=(x²+2x+2)−x²=2x+2.
- Both sides agree for all real x, so the equation holds.
- At x=0 the candidate equals 2, so it also satisfies the initial condition.
Passing the equation test does not automatically pass the initial-value test. Matching one sampled point does not verify a solution.
Is y=2e^x+1 a solution of y′=y?
Compare with an explanation
No. Its derivative is 2e^x, which differs from y by −1.
Predict. Change one thing. Explain.
Switch candidates for y′=y, y(0)=2. Compare the equation residual with the initial value. Explain why 3e^x passes one test and fails the other.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
Equation holds algebraically; initial value is 2, matching required 2. Navy y; orange dashed y′.
Two independent checks
Residual y′−y: identically 0.
At x=0: y=2.
Both requirements hold.
Agreement of plotted samples is not an identity proof.
Original model; rounded readouts and finite sampled graphs. Equation y′=y with initial condition y(0)=2. Axes dimensionless. A sampled residual is illustrative; the algebra checks the identity.
Explain what you noticed: Answer the investigation prompt above. State one observation and explain it using the rate units, derivative signs, initial condition, domain or solution check. Identify what the representation cannot tell you.
Apply the idea to a fresh problem Practice →Show what you understand.
Two original questions are a starting check, not proof of mastery. Explain your choice before revealing the answer.
Original written challenge
4 points · self-check · not an official AP questionVerify y=4e^(−2t) for y′=−2y, y(0)=4, and explain why 5e^(−2t) does not solve the same initial-value problem.
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Compare with the answer and four-point rubric
- 1 point: Differentiate to −8e^(−2t).
- 1 point: Compute −2y=−8e^(−2t), equal for all real t.
- 1 point: Evaluate y(0)=4.
- 1 point: The second candidate satisfies the equation but starts at 5 instead of 4.
Accept equivalent correct methods and explanations. This is a Refresh Kid teaching rubric, not an official AP scoring guideline.
Retrieve it before you reveal it.
RECALL 1What is a residual?
The substituted left side minus right side.
RECALL 2What two tests define verification of an IVP?
The differential equation on an interval and the initial condition.
RECALL 3Can a differential equation have many solutions?
Yes; for example y′=y has the family Ce^x.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
How can you check a proposed solution without solving the equation?
- Check y′(x)=F(x,y(x)) on an interval.
- Check y(a)=b separately.
Remember: Passing the equation test does not automatically pass the initial-value test. Matching one sampled point does not verify a solution.
Conditions: Original model; rounded readouts and finite sampled graphs. Equation y′=y with initial condition y(0)=2. Axes dimensionless. A sampled residual is illustrative; the algebra checks the identity.
Refresh Kid · AP Calculus BC Unit 7 · Objectives FUN-7.B · Review edition
Framework, scope and review status
Mapped to College Board CED Topic 7.2, FUN-7.B. CED and Fall 2026 clarifications checked September 17, 2026. All nine BC topics are included, with Euler’s method and logistic interpretation. The correction to FUN-7.B.2 is reflected: an equation may have infinitely many solutions. Focused explanations, examples, visual models and practice are original Refresh Kid work.
Rate and amount units, signs, initial data, lost equilibria, solution intervals and model assumptions are explicit. Slope fields and Euler polygons are finite illustrations; exact algebra supports identities and limits. The logistic explicit formula supports the visualization; required interpretation can be done from the rate law without deriving that formula. The smooth examples have unique solutions; no universal uniqueness claim is made.
Organic Chemistry Tutor video titles and destinations were checked, not the full videos. Khan Academy’s destination was checked; its JavaScript lesson contents were not fully readable by the research tool. OpenStax sections 4.1–4.4 supplied conceptual cross-checks. No provider questions, artwork or scripts were copied. No affiliation or endorsement is implied.
GitHub’s 3D website collection informed optional spatial inspection and camera controls. The original tank reuses self-hosted Three.js with its MIT license retained. The 2×2 dm base connects water depth to volume: 1 dm³=1 L. This tank uses a feedback-controlled pump, not a gravity-drain law. Its 3D height and labeled 2D graphs use the same exact exponential formula. Rotation changes only the view, and complete explanations remain available without WebGL.
Independent teacher review and student usability testing remain pending. Technical checks do not certify mathematical accuracy, full accessibility or learning effectiveness. This is a review edition.
Optional official resource: Released AP Calculus BC questions and scoring guides. This multi-unit archive is not assigned as a complete Unit 7 task.
Learn → Explore → Practice → Review is informed by the IES learning guide; this exact implementation has not been evaluated with learners.
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