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Integrating Factors for Linear Equations 

On this page, we will look at integrating factors for firstorder, linear equations. (See the exact differential equations page for discussion of integrating factors for inexact equations. ) The integrating factor converts the differential equation into a form that can be solved by direct integration. 
The integrating factor we will look at applies to firstorder, linear differential equations of the form
\(\displaystyle{ \frac{dy}{dt} + p(t)y = g(t) }\).
The idea of the technique of integrating factors is deceptively simple, yet quite powerful. When you have a firstorder, linear differential equation of the form
\(\displaystyle{ \frac{dy}{dt} + p(t)y = g(t) }\)
and you multiply this equation by the generated integrating factor
\( \mu(t) = \exp \int{ p(t)~dt }\)
[ what does exp mean? ]
this converts the differential equation into the form
\(\displaystyle{ \frac{d}{dt}[ \mu(t)y] = \mu(t)g(t) }\)
You can then integrate to get \(\displaystyle{ \mu(t)y }\) and divide by \(\mu(t)\) to solve for \(y\).
[ Note: We will not go through the derivation of the integrating factor here at this time. However, going through the derivation in your textbook will really help you understand what is going on here. ]
Okay, let's watch some videos, so we can see how this works.
Here is a good introduction to integrating factors that is not too long.
MIT OCW  integrating factors introduction  
Here are a couple of complete examples using integrating factors to solve firstorder, linear differential equations.
Dr Chris Tisdell  integrating factors  
Okay, time for some practice problems.
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Practice Problems 

Instructions   Unless otherwise instructed, find the general solutions to these differential equations using the method of integrating factors. If initial condition(s) are given, find the particular solution also. Give your answers in exact form.