5. First Order Linear Differential Equations

First Order Linear Differential Equations

Definition

A first-order linear differential equation is one that can be put into the form

$\frac{dy}{dx}+P(x)y=Q(x)$

We call the equation above the standard form.

Integrating Factor

Definition

The integrating factor $I(x)$ for the linear differential equation (in standard form)

$\frac{dy}{dx}+P(x)y=Q(x)$

is the function

$I(x)=e^{\int P(x)dx}$

where you may choose any value for the integration constant.

Purpose

An integrating factor $I(x)=e^{\int P(x)dx}$ for the linear differential equation

$\frac{dy}{dx}+P(x)y=Q(x)$

satisfies

$I^{\prime }(x)=I(x)P(x)$

Corollary

$[I(x)y{]}^{\prime }=I(x)y^{\prime }+I^{\prime }(x)y=I(x)y^{\prime }+I(x)P(x)y=I(x)Q(x)$

Solution Procedure - Algorithm

Algorithm to solve 1st order linear differential equations:

1. Write in standard form: $\frac{dy}{dx}+P(x)y=Q(x)$
2. Compute integrating factor: $I(x)=e^{\int P(x)dx}$
3. Multiply by $I(x)$: $[I(x)y{]}^{\prime }=I(x)y^{\prime }+I(x)P(x)y=I(x)Q(x)$
4. Integrate: $I(x)y=\int I(x)Q(x)dx$
5. Divide by $I(x)$: $y(x)=\frac{1}{I(x)}\int I(x)Q(x)dx$