Example: Solve the initial value problem
Answer: Notice that for x=1, we have . 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
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
We get . The non-constant solutions may be obtained by first separating
and then performing the integration
The technique of integration of rational functions gives
If we set y=2 when t=0, we get
Easy algebraic manipulations give
Therefore, the solution Y = (x,y), where
satisfies the initial condition . By the existence and uniqueness theorem, this is the desired solution.
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Author: Mohamed Amine Khamsi