Linear Equations: Answer to Example 1

Example: Find the solution to

displaymath73.


Solution: First, we recognize that this is a linear equation. Indeed, we have

displaymath75

Therefore, the integrating factor is given by

displaymath77.

Since tex2html_wrap_inline79 , we get

displaymath81

Hence, the general solution is given by the formula

displaymath83

We have

displaymath85

The details for this calculation involve the technique of integrating rational functions. We have

displaymath87

Hence, the only difficulty is in the integral tex2html_wrap_inline89 . Here we will use integration by parts. We will differentiate t and integrate tex2html_wrap_inline93 . The details are left to the reader. We have

displaymath95

Therefore, we have

displaymath97,

which clearly implies

displaymath99

The general solution can also be rewritten as

displaymath101

Finally, the initial condition y(0) = 0.4 gives C = 0.4. Therefore, the solution to the IVP is

displaymath107

Next Example.

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Author: Mohamed Amine Khamsi
Last Update 6-22-98

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