Answer to Example 5

Example: Find the solution to the system


under the initial condition tex2html_wrap_inline52 .

Answer: Notice that the second equation of the system is a differential equation involving only the variable y. Its integration gives


Note that tex2html_wrap_inline58 is the derivative of tex2html_wrap_inline60 not its anti-derivative! The initial condition tex2html_wrap_inline52 translates into the initial condition y(0)=0 for the variable y. Hence, we have


which gives tex2html_wrap_inline70 . Since we have y, we plug it into the first equation to get


We recognize a first order linear differential equation. In order to solve it, first we need to find the integrating factor given by


Note that the anti-derivative of tex2html_wrap_inline78 is tex2html_wrap_inline80 . The general solution is then given by


We have




where, in the first integral, we used direct tables and for the second one we used integration by parts (we integrated tex2html_wrap_inline60 and differentiated t). Putting everything together, we get


The initial condition tex2html_wrap_inline52 translates into the initial condition x(0)=1 for the variable x. Hence, we have


which gives C=0. Therefore, we have


Finally, the solution to the system is


Note that since tex2html_wrap_inline108 , we may generate another expression for the function x(t).

Next Example:

[Differential Equations] [First Order D.E.]
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Author: Mohamed Amine Khamsi

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