General Solution Method for

1st 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 1st order linear DE can always be written in the form

 

or rearranged to the form .

The steps to solve such equations are summarized below.

 

  1. Put the equation in the standard form. (The equation with P(x) and Q(x) above.)
  2. Obtain the integrating factor

  3. Multiply the standard equation by the integrating factor.
  4. The left-hand side of the equation should simplify to a derivative of the product of the integrating factor and the dependent variable.
  5. Solve the equation by integrating, using the initial conditions if necessary.