Answer to Example 6

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

,

which implies

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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