First order linear differential equation

A first order linear differential equation has the following form:

The general solution is given by

where

called the integrating factor. If an initial condition is given, use it to find the constant C.

If the differential equation is given as

rewrite it in the form

where

Find the integrating factor

Evaluate the integral and write down the general solution

If you are given an IVP, use the initial condition to find the constant C.

Example 1

Consider the differential equation y'(x) + 2y(x) = 6.

The general solution of this equation is y(x) = Ce^-2x + 3.

For the initial condition y(0) = 10, we obtain C = 7, so that the solution is y(x) = 7e^-2x + 3.

This solution is stable, because 2 > 0.

Example 2

When the price of a good is p, the total demand is D(p) = a - bp and the total supply is S(p) = α + βp, where a, b, α, and β are positive constants. When demand exceeds supply, price rises, and when supply exceeds demand it falls. The speed at which the price changes is proportional to the difference between supply and demand. Specifically,

p'(t) = λ[D(p) - S(p)]

with λ > 0. Given the forms of the supply and demand functions, we thus have

p'(t) + λ(b + β)p(t) = λ(a - α).

The general solution of this differential equation is

p(t) = Ce^{-λ(b + β)t} + (a - α)/(b + β).

The equilibrium price is (a - α)/(b + β), and, given λ(b + β) > 0, this equilibrium is globally stable.