Introduction: Answer to Example3
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 
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.
S.O.S MATHematics home page Do you need more help? Please post your question on our S.O.S. Mathematics CyberBoard..
Author: Mohamed Amine Khamsi