**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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