Answer to Example 5

Example: Find the solution to the system

displaymath50

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

displaymath56

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

displaymath68,

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

displaymath74

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

displaymath76

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

displaymath82

We have

displaymath84,

and

displaymath86,

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

displaymath92.

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

displaymath100

which gives C=0. Therefore, we have

displaymath104.

Finally, the solution to the system is

displaymath106

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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