Before we give the formal definition of **Riccati equations**,
a little introduction may be helpful.
Indeed, consider the first order differential equation

If we approximate

If we stop at

These equations bear his name,

Consider the new function

Then easy calculations give

which is a linear equation satisfied by the new function

Keep in mind that it may be harder to remember the above equation satisfied by

**Example.** Solve the equation

knowing that

**Answer.** We recognize a Riccati equation. First of all we need to make sure
that *y*_{1} is indeed a solution. Otherwise, our calculations will be fruitless.
In this particular case, it is quite easy to check that *y*_{1} = 2 is a solution. Set

Then we have

which implies

Hence, from the equation satisfied by

Easy algebraic manipulations give

Hence

This is a linear equation. The general solution is given by

Therefore, we have

Note: If one remembers the equation satisfied by *z*,
then the solutions may be found a bit faster. Indeed in this example, we have *P*(*x*) = -2, *Q*(*x*) = -1, and *R*(*x*) = 1. Hence the linear equation satisfied by the new function *z*, is

**Example.** Check that
is a solution to

Then solve the IVP

We will let the reader check that is indeed a particular solution of the given differential equations. We also recognize that the equation is of Riccati type. Set

which gives

Hence

Substituting into the equation gives

Easy algebraic manipulations give

Hence

This is the linear equation satisfied by

The general solution is

Now it is time to go back to the original function

The initial condition

**
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