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
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 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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Author: Mohamed Amine Khamsi