Solving
Even though ‘contrib_ode’, Maxima’s ODE solver choked on this equation (see “An Alternate Solver of ODEs“), it still can be solved as demonstrated below:
multiplied by
i.e.,
Integrate it, we have
By (see “Integration by Parts Done Right“),
As a result, (1-1) yields a new ODE
or,
Upon submitting (1-3) to Omega CAS Explorer in non-expert mode, the CAS asks for the range of
Fig. 1
Depending on the range provided, ‘ode2’ gives three different solutions (see Fig. 2, 3 and 4).
Fig. 2
Fig. 3
Fig. 4
Let’s also solve (1-3) manually:
If (1-3) has a constant solution
In fact, this solution can be observed from right away.
Otherwise (),
That is,
or
For let
we have
Divide both numerator and denominator on the left side by ,
Write it as
Multiply both sides by
Integrate it,
we obtain
i.e.,
or,
where
For let
For
Notice when (1-2) becomes
This is a Bernoulli’s Equation with
and
Solving it (see “Meeting Mr. Bernoulli“),
Since we can verify (3-1) as follows:
substitute into (2-3),
Unsurprisingly, this is the same as (3-1).
Exercise-1 Mathematica solves

But it only return one solution. Show that it is equivalent to (2-2).
Exercise-2 Solving (1-2) using ‘contrib_ode’.
Exercise-3 Show that (2-2), (2-3) and (2-4) are equivalent to results shown in Fig. 2, 3 and 4 respectively.