Introduction: Answer to Example3

Example: Find the solution to the system


Answer: First, solve the second equation since it does not contain the variable x. We recognize a separable equation. Hence, we will first look for the constant solutions.


This clearly gives y=2. The non-constant solutions can be obtained by separating the variables


and then performing integration. Since


we get


If we put all the solutions together we get


Clearly, the only solution satisfying the initial condition y(0)=2 is the constant solution y=2. Next, we plug the value of y(t) into the first equation of the system to get


This again is a separable equation. This time we do not have constant solutions since the quadratic equation tex2html_wrap_inline101 does not have real roots. Let us find the non-constant solutions. First, we separate the variables x and t to get


Since we have (using the techniques of integration of rational functions)


then we get


The initial condition x(0)=0 gives


Finally, the solution to the system is


You may want to find x explicitly as a function of t.

Remark: Since the constant solution y=2 is the solution of the second equation and the initial condition to be satisfied by y is y(0)=2, we may conclude directly from existence and uniqueness, that y=2 is the desired solution without looking for the non-constant solutions.

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