Answer to Example 6

Example: Solve the initial value problem

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Answer: Notice that for x=1, we have tex2html_wrap_inline71 . Hence, the constant function x(t) = 1 is solution to the first equation of the system. Set x=1 in the second equation to get

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This is a first order differential equation which is separable. Let us solve it. First, we look for the constant solutions which may be obtained from

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We get tex2html_wrap_inline81 . The non-constant solutions may be obtained by first separating

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and then performing the integration

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The technique of integration of rational functions gives

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which implies

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If we set y=2 when t=0, we get

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Easy algebraic manipulations give

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Therefore, the solution Y = (x,y), where

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satisfies the initial condition tex2html_wrap_inline103 . By the existence and uniqueness theorem, this is the desired solution.

[Differential Equations] [First Order D.E.]
[Geometry] [Algebra] [Trigonometry ]
[Calculus] [Complex Variables] [Matrix Algebra]

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Author: Mohamed Amine Khamsi

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