Finding Particular Solutions Using Initial Conditions and Separation of Variables
Use an initial condition to choose a solution, branch, and connected interval of validity.
College Board Topic 7.7 · CED p. 138 (PDF p. 143) · CED effective Fall 2020; project course outline 2026–27.
A general solution describes a family. Apply the initial condition to determine its constant and select the appropriate branch and interval. For the smooth separated models studied here, a valid initial condition selects one curve from that family. Verify the equation and the condition rather than assuming that all initial-value problems are automatically unique.
If y′=f(x) and y(a)=y₀, a particular solution is y(x)=y₀+∫ₐˣ f(t)dt. This works even without an elementary antiderivative, provided f is continuous. With y′=y² and y(0)=1, separation gives y=1/(1−x). The interval containing 0 is x<1; the formula has a vertical asymptote at x=1 and cannot describe one continuous solution through that point. An algebraic expression and its solution domain belong together.
Let's work through it
- For y′=xy and y(0)=2, use general family y=Ae^(x²/2). Reason: Separate variables and integrate.
- Substitute 2=Ae⁰, giving A=2. Reason: The initial condition determines the family member.
- Thus y=2e^(x²/2), valid for all real x, and its derivative is xy. Reason: There are no singularities, and direct substitution verifies the result.
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
If y′=f(x), y(3)=7, what is the accumulation form?
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Retrieval 1
If y′=f(x), y(3)=7, what is the accumulation form?
Worked answer and reason
y(x)=7+∫₃ˣ f(t)dt.
The integral gives change from the initial point.
Retrieval 2
Why give an interval along with y=1/(1−x), y(0)=1?
Worked answer and reason
The solution cannot pass through x=1.
The denominator vanishes there; use the connected interval containing the initial point.
Catch the tempting mistake
- Solving for C finishes the problem even if the expression crosses a pole.
- State the solution interval containing the initial condition and verify the original equation on it.
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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Put it to work
- Unit 7 problem 17 · No calculator · analytic
- Unit 7 problem 18 · No calculator · analytic
- Unit 7 problem 19 · No calculator · analytic
- Unit 7 problem 20 · No calculator · analytic
- Unit 7 problem 21 · No calculator · analytic
For the teacher
- Include one initial condition that sets an equilibrium and one that requires a restricted interval. Ask students to compare explicit and accumulation forms.
Glossary and deeper reading · 4 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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General solution
A general solution describes a family of functions satisfying a differential equation. Arbitrary constants account for the freedom left by the equation. For example, y=x²+C is a general solution of y′=2x. Separation or other manipulations can miss special solutions, so the family and any excluded cases need checking.
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Initial condition
An initial condition specifies a solution’s value at a particular input. It can determine the constant in a family of solutions to a differential equation. For example, y′=2x with y(0)=3 selects y=x²+3. The solution must satisfy both the equation and the stated condition on a suitable interval.
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Antiderivative
An antiderivative of a function is another function whose derivative equals the original function on an interval. Two antiderivatives on the same interval differ by a constant. For example, x² is an antiderivative of 2x, and x²+7 is another. An antiderivative is a function, whereas a definite integral is an accumulated value.
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Interval
An interval is a set of real numbers containing every number between any two of its members. Parentheses exclude an endpoint, while square brackets include it. For example, (1,4] includes 4 and every number between 1 and 4, but excludes 1. An infinite endpoint always uses a parenthesis because infinity is not a real endpoint.