First Order Homogeneous Linear Differential Equation

(1)

y + p(t)y = 0.

It is understood that t varies over some interval in the real line, and p(t) is a continuous function of t in the interval. The equation is called linear because y and y' occur only linearly and homogeneous because the right side of the equation is zero. The equation

y' = ky, or y' - ky = 0,

for exponential growth (see Section 8.6) is an example. The first order homogeneous linear differential equation (1) has separable variables, because it can be written as

Its solution is given by the next formula.