**Contents:**

What is a Particular Solution?

Find particular solution: Example

## What is a Particular Solution?

A **particular solution** requires you to find a single solution that meets the constraints of the question. A problem that asks you to find a **series of functions** has a general solution as the answer—a solution that contains a constant (+ C), which could represent one of a possibly infinite number of functions.

For example, a problem with the differential equation

^{dy}⁄_{dv} x^{3} + 8

requires a general solution with a constant for the answer, while the differential equation

^{dy}⁄_{dv} x^{3} + 8; f(0) = 2

requires a particular solution, one that fits the constraint f(0) = 2.

Watch this 5 minute video showing the difference between particular and general, or read on below for how to find particular solution differential equations.

## Find Particular solution: Example

**Example problem #1: **Find the particular solution for the differential equation ^{dy}⁄_{dx} = 5, where y(0) = 2.

Step 1: **Rewrite the equation using algebra** to move dx to the right (this step makes integration possible):

- dy = 5 dx

Step 2: **Integrate both sides of the equation** to get the general solution differential equation. Need to brush up on the rules? See: Common integration rules.

- ∫ dy = ∫ 5 dx →
- ∫ 1 dy = ∫ 5 dx →
- y = 5x + C

Step 3: **Rewrite the general equation** to satisfy the **initial condition**, which stated that when x = 0, y = 2:

- 2 = 5(0) + C
- C = 2

The differential equation particular solution is y = 5x + 2

**Particular solution differential equations, Example problem #2: **

Find the particular solution for the differential equation ^{dy}⁄_{dx}= 18x, where y(5) = 230.

Step 1: **Rewrite the equation using algebra** to move dx to the right:

- dy = 18x dx

Step 2: **Integrate both sides of the equation**:

- ∫ dy = ∫ 18x dx →
- ∫ 1 dy = ∫ 18x dx →
- y = 9x
^{2}+ C

Step 3: **Rewrite the general equation** to satisfy the **initial condition**, which stated that when x = 5, y = 230:

- 230 = 9(5)
^{2}+ C - C = 5

The differential equation particular solution is y = 5x + 5

*That’s it!*

## References

4.5 The Superposition Principle and Undetermined Coefficients Revisited.