A first order linear differential equation is an equation that relates an unknown function with its first derivative. It can be written in the form of y' + p(x)y = q(x), where p(x) and q(x) are functions of x.
The general solution to a first order linear differential equation can be found by using an integrating factor, which is a function that multiplies both sides of the equation to make it easier to solve. The integrating factor is e∫p(x)dx, and once found, the solution is given by y = 1/IF * ∫IF * q(x)dx + c, where IF is the integrating factor and c is the constant of integration.
A first order linear differential equation has a constant coefficient for the highest order derivative, while a first order non-linear differential equation has a variable coefficient for the highest order derivative. This means that the unknown function and its derivatives are linearly related in a first order linear differential equation, while they are non-linearly related in a first order non-linear differential equation.
To apply initial conditions, you simply substitute the given values of the unknown function and its derivative into the general solution. This will give you a specific solution that satisfies the given initial conditions.
First order linear differential equations are commonly used in physics and engineering to model processes such as growth, decay, and heat transfer. They are also used in economics to model population growth and interest rates, and in biology to model population dynamics and chemical reactions. Many other fields also use first order linear differential equations to describe various phenomena.