Finding General Solutions Using Separation of Variables
Separate variable factors, check lost equilibria, and verify the resulting solution family.
College Board Topic 7.6 · CED p. 137 (PDF p. 142) · CED effective Fall 2020; project course outline 2026–27.
If y′=g(x)h(y), move the y-dependent factor to one side and the x-dependent factor to the other: dy/h(y)=g(x)dx. Integrate both sides, then solve for y when possible. The constant records a family of solutions. You are using the chain rule in reverse, so check by differentiating the final equation or expression.
Dividing by h(y) assumes h(y)≠0. Check the zeros of h first: they may give equilibrium solutions that division would lose. For y′=xy, separation for y≠0 gives ln|y|=x²/2+C and y=Ae^(x²/2), with A allowed to be positive, negative, or zero when the equilibrium y=0 is included. Not every equation separates: y′=x+y is not a product of an x-only and a y-only function. Keep any domain restrictions from logarithms, radicals, or denominators.
Let's work through it
- Solve y′=2xy for y≠0: dy/y=2x dx. Reason: The equation factors into an x-part and a y-part.
- Integrate to ln|y|=x²+C and rewrite y=Ae^(x²). Reason: Antidifferentiation and exponentiation give a family; A carries the sign.
- Include y=0, represented by A=0. Reason: It solves the original equation but was excluded during division.
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
What must be checked before dividing by y?
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
What must be checked before dividing by y?
Worked answer and reason
The possible solution y=0.
Division by y excludes zero and can lose an equilibrium.
Retrieval 2
Is y′=x+y immediately separable?
Worked answer and reason
No.
A sum cannot be split into an x-only factor times a y-only factor by that rearrangement.
Catch the tempting mistake
- Exponentiating ln|y| forces y to be positive.
- The absolute value permits positive or negative solutions on their intervals; include the sign in the constant A.
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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+ − * / ^ ! · pi · e · sqrt · sin, cos, tan · asin, acos, atan · atan2(y,x) · ln, log · abs · exp · hypot(x,y). Use * between factors. Keep units in your written working.
Put it to work
- Unit 7 problem 15 · No calculator · analytic
- Unit 7 problem 16 · No calculator · analytic
- Unit 7 problem 17 · No calculator · analytic
For the teacher
- Require an equilibrium check before separation and a derivative check afterward. Compare a separable equation with a similar nonseparable one.
Glossary and deeper reading · 3 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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Chain rule
The chain rule differentiates a composition by multiplying the outer derivative at the inner output by the inner derivative. For differentiable functions, (f(g(x)))′=f′(g(x))g′(x). For example, the derivative of sin(x²) is cos(x²)·2x. Nested compositions require a factor for each successive layer.
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Function
A mathematical function assigns exactly one output to each input in its domain. Inputs can share an output, but one input cannot have two different outputs. For example, f(x)=x² assigns the output 9 to both 3 and −3. Function notation names the rule and its input; f(3) means the output when the input is 3.
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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.