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:
Integrate it, we have
By (see “Integration by Parts Done Right“),
As a result, (1-1) yields a new ODE
Upon submitting (1-3) to Omega CAS Explorer in non-expert mode, the CAS asks for the range of
Depending on the range provided, ‘ode2’ gives three different solutions (see Fig. 2, 3 and 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.
For let we have
Divide both numerator and denominator on the left side by ,
Write it as
Multiply both sides by
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.