The following transformation
leads to another Riccati ODE, but with h(x) -> 1:
The following transformation
leads to another Riccati ODE but without the "linear in u(t)" term:
Depending on the case, the system might be able to solve the problem using one of the transformations above; the above transformations can also be used as departure point for guessing another transformation suitable for the given problem. It is also possible to introduce a change of variables leading to a linear ODE of the second order (see below).
Concerning Riccati ODEs of Special type, the solving scheme can be summarized as follows. The first thing worth noting is that, for the system succeeds in solving the ODE:
Now, for , Riccati Special ODEs can be reduced, step by step, to the case , provided that can be written as (integer ). Examples of possible values for :
The idea is to change variables so as to obtain another ODE of type Riccati Special, but with c[n->0], until reaching . There are two variable transformations leading to the desired reduction, depending on the sign of in above (see examples at the end).
1) Riccati Special ODE with ()
The general transformation for n > 0 (in ), in order to reduce is given by
In each step, you must introduce the corresponding value of before using it.
The change of variables plus a few simplifications leads to the desired new_ode of type Riccati Special with the order of reduced by one.
The exponent of was reduced from to . Keep doing that until the value is reached. Then, as shown above, a solution is already known.
The same idea applies when n < 0 (in c[n]). For this case, the change of variables is given by
The case in which f(x) + g(x) + h(x) = 0
Finally, it is also possible to convert Riccati ODEs in second order linear homogeneous ODEs, by using convert,ODEs), as follows:
In the above, the first operand is the second order linear ODE and the second operand is the transformation of variables used.
