# Differential Equations

# More About Solutions

Differential equations have two kinds of solutions: **general** and **particular**.

The **general solution** to a differential equation is the collection of all solutions to that differential equation. A general solution will usually contain some undetermined constants.

### Sample Problem

*y* = 4*x* + *C* is the general solution to the d.e.

*y *' = 4.

### Sample Problem

*y* = 3*x*^{2} + *Bx* + *C* is the general solution to the d.e.

*y *" = 6.

A **particular solution** to a differential equation is a solution with all the constants filled in.

### Sample Problem

The function *y* = 4*x* + 2 is a specific solution to the d.e.

*y *' = 4.

Initial value problems usually have a particular solution only, because the initial condition(s) force us to pick values for the constant(s) in the general solution.