First-Order Linear Differential Equations

DEFINITION. The general first-order linear differential equation is

                                                                 (1)

Unless otherwise stated, the functions and are assumed to be continuous functions of time. The equation

                                                                   (2)

is called the homogeneous first-order linear differential equation, and Equation (1) is called the nonhomogeneous first-order linear differential equation for not identically zero.

Fortunately, the homogeneous equation (2) can be solved quite easily. One can show that the general solution of (2)

                                                          (3)

where is any real number.

EXAMPLE: Find the general solution of the equation

                                                                    (4)



In applications, we are usually not interested in all solutions of (2). Rather, we are looking for the specific solution which at some initial time has the value . Thus, we want to determine a function such that



One can show that

                                                        (5)

EXAMPLE: Find the solution of the initial-value problem




EXAMPLE: Find the solution of the initial-value problem




We return now to the nonhomogeneous equation



One can show that



where




Alternately, if we are interested in the specific solution of (1) satisfying the initial condition , that is, if we want to solve the initial-value problem



then we can take the definite integral of both sides of (7) between and to obtain that



or



EXAMPLE: Find the general solution of the equation

                                                                     (8)



EXAMPLE: Find the solution of the initial-value problem

                                                        (9)



EXAMPLE: Find the solution of the initial-value problem

                                                  (10)



EXAMPLE: Solve