General Solution Method for

1^{st} Order Linear Differential Equations

A differential equation is linear if it is a linear function of x, y, y’, y"…and the order is the order of the highest derivative.

A 1^{st} order linear DE can always be written in the form

or rearranged to the form .

The steps to solve such equations are summarized below.

- Put the equation in the standard form. (The equation with P(x) and Q(x) above.)
- Obtain the integrating factor
- Multiply the standard equation by the integrating factor.
- The left-hand side of the equation should simplify to a derivative of the product of the integrating factor and the dependent variable.
- Solve the equation by integrating, using the initial conditions if necessary.