Particular Solution — Definition, Formula & Examples

A particular solution is the single specific solution you get from a differential equation's general solution after applying an initial condition or boundary condition to determine the unknown constant(s).

Given a differential equation with general solution $y = f(x, C_1, C_2, \ldots, C_n)$ containing $n$ arbitrary constants, a particular solution is obtained by assigning definite values to each constant so that the solution satisfies prescribed conditions such as $y(x_0) = y_0$ .

How It Works

Start by solving the differential equation to find its general solution, which includes one or more arbitrary constants. Then substitute the given initial or boundary conditions into the general solution. Solve the resulting equation(s) for the constant(s). Replace the constant(s) in the general solution with their computed values to write the particular solution.

Worked Example

Problem: Find the particular solution of

\frac{dy}{dx} = 2x

given that

y(1) = 5

Find the general solution: Integrate both sides with respect to

x

y = \int 2x\,dx = x^2 + C

Apply the initial condition: Substitute

x = 1

and

y = 5

into the general solution to find

C

5 = (1)^2 + C \implies C = 4

Write the particular solution: Replace

C

with 4 in the general solution.

y = x^2 + 4

Answer: The particular solution is

y = x^2 + 4

Why It Matters

In AP Calculus BC Unit 7, you must solve initial value problems, which means finding particular solutions. Real-world models—population growth, cooling rates, velocity functions—require a particular solution to make a unique prediction rather than describing a family of curves.

Common Mistakes

Mistake: Confusing the general solution with the particular solution and leaving arbitrary constants in the final answer.

Correction: A particular solution contains no arbitrary constants. Always apply the given condition to solve for every constant before writing your final answer.

Related Terms

Initial Value Problem — Provides the condition that determines a particular solution
Differential Equation — The equation a particular solution satisfies
First Order Differential Equation — Common type yielding one constant to solve for
Boundary Value Problem (BVP) — Alternative condition type that fixes constants
Second Order Differential Equation — Requires two conditions for a particular solution
Order of a Differential Equation — Determines how many constants appear in the general solution