How do you separate variables to find a solution family?
You will be able to: Separate a product rate law and integrate both sides with a constant.
How do you separate variables to find a solution family?
Suppose a rate depends on both elapsed input x and the amount y: y′=2xy. Knowing only the rate rule leaves many possible starting amounts. Separation organizes the equation so each side contains one variable.
A useful starting point: How do step size and direction affect Euler’s method? →
Words and symbols before equations
- Separable equation
- An equation y′=f(x)g(y) whose variables can be collected on opposite sides.
- Antiderivative
- A function whose derivative is the integrand.
- Integration constant
- A free constant describing a family.
- Equilibrium check
- Testing values excluded when dividing by g(y).
What this picture assumes
Original model; rounded readouts and finite sampled graphs. y′=2xy, family y=Ae^(x²), x in[−1,1]. A=0 restores the equilibrium excluded by division. Displayed variables are dimensionless.
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.
- y=1e^(x²−0), passing through (0,1). Differentiating gives y′=2xy.
- Check what the picture assumes below. Use the Explore task to predict one change before moving a control.
Connect the picture to the mathematics
For y≠0, divide by y and write dy/y=2x dx. Integrate to ln∣y∣=x²+C. One constant suffices because the difference of two arbitrary constants is another arbitrary constant.
Exponentiating yields ∣y∣=e^C e^(x²). The sign of y is constant on any nonzero solution interval, so combine sign and e^C into A to get y=Ae^(x²), A≠0.
Check y=0 directly in the original equation: it also works. Allowing A=0 includes this lost equilibrium in the family.
Separation requires a product f(x)g(y); a sum such as x+y cannot be separated by writing dy/y=x dx. Derivative verification catches invalid rearrangements.
A worked example, step by step
Find the family of solutions to y′=3y, including the equilibrium.
- For y≠0, separate dy/y=3 dx.
- Integrate ln∣y∣=3x+C.
- Exponentiate to y=Ae^(3x) for nonzero A.
- Check y=0 separately; including A=0 gives the whole family, and differentiation verifies y′=3y.
Do not drop absolute values during logarithmic integration without handling sign. Check every value excluded by division.
Why is one arbitrary constant enough?
Compare with an explanation
Constants from the two integrations can be combined into a single arbitrary difference.
Predict. Change one thing. Explain.
Move A through negative values, zero and positive values. Identify the equilibrium and explain how all these curves satisfy y′=2xy although their signs differ.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
y=1e^(x²−0), passing through (0,1). Differentiating gives y′=2xy.
Derivation and check
For y≠0: dy/y=2x dx.
ln|y|=x²+C; y=Ae^(x²).
The chosen coefficient A=1.
All real x are valid; the displayed window is finite.
Original model; rounded readouts and finite sampled graphs. y′=2xy, family y=Ae^(x²), x in[−1,1]. A=0 restores the equilibrium excluded by division. Displayed variables are dimensionless.
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 questionSolve y′=4xy as a general family and verify your answer.
This response is not submitted or saved. Copy it before leaving.
Compare with the answer and four-point rubric
- 1 point: For y≠0, separate dy/y=4x dx.
- 1 point: Integrate ln∣y∣=2x²+C.
- 1 point: Obtain y=Ae^(2x²); include A=0 after checking the equilibrium.
- 1 point: Differentiate to 4xAe^(2x²)=4xy.
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 1When is separation valid?
When the rate can be written as a product f(x)g(y), on a domain where the division is allowed.
RECALL 2Why keep an arbitrary constant?
The rate equation alone may permit many initial values.
RECALL 3How do you check the final family?
Differentiate and substitute, including equilibrium cases.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
How do you separate variables to find a solution family?
- dy/g(y)=f(x)dx when g(y)≠0.
- For y′=2xy: y=Ae^(x²), including A=0.
Remember: Do not drop absolute values during logarithmic integration without handling sign. Check every value excluded by division.
Conditions: Original model; rounded readouts and finite sampled graphs. y′=2xy, family y=Ae^(x²), x in[−1,1]. A=0 restores the equilibrium excluded by division. Displayed variables are dimensionless.
Refresh Kid · AP Calculus BC Unit 7 · Objectives FUN-7.D · Review edition
Framework, scope and review status
Mapped to College Board CED Topic 7.6, FUN-7.D. 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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