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.