Linear Differential Equations

Posted By on January 3, 2015


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Separable Equations

A first order linear differential equation has the following form:

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The general solution is given by

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where

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

Here are some practical steps to follow:

1.
If the differential equation is given asdisplaymath49,

rewrite it in the form

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where

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2.
Find the integrating factordisplaymath53.

3.
Evaluate the integral tex2html_wrap_inline55
4.
Write down the general solutiondisplaymath57.

5.
If you are given an IVP, use the initial condition to find the constant C.

Example: Find the particular solution of:

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Solution: Let us use the steps:

Step 1: There is no need for rewriting the differential equation. We havedisplaymath63

Step 2: Integrating factordisplaymath65.

Step 3: We havedisplaymath67.

Step 4: The general solution is given bydisplaymath69.

Step 5: In order to find the particular solution to the given IVP, we use the initial condition to find C. Indeed, we havedisplaymath73.

Therefore the solution is

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Note that you may not have to do the last step if you are asked to find the general solution (not an IVP).

Separable Equations

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Posted by Akash Kurup

Founder and C.E.O, World4Engineers Educationist and Entrepreneur by passion. Orator and blogger by hobby

Website: http://world4engineers.com