# 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.

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