Answer:
See solution below
Step-by-step explanation:
(a) If n=0 or 1, the equation
(1)
would be a simple linear differential equation. So, we can assume that n is different to 0 or 1.
Let's use the following substitution:
(2)
Taking the derivative implicitly and using the chain rule:
(3)
Multiplying equation (1) on both sides by
we obtain the equation
reordering:
Now, using (2) and (3) we get:
which is an ordinary linear differential equation with unknown function z(t).
(b)
The equation we want to solve is
(4)
Here, our independent variable is x (instead of t)
Assuming x different to 0, we divide both sides by x to obtain:
Which is an equation of the form (1) with
So, if we substitute
we transform equation (4) in the lineal equation
(5)
and this is an ordinary lineal differential equation of first order whose
integrating factor is
but
Similarly,
and the general solution of (5) is then
where C is any real constant
Reversing the substitution
we obtain the general solution of (4)
Attached there is a sketch of several particular solutions corresponding to C=1,4,6
It is worth noticing that the solutions are not defined on x=0 and for C<0
