problem: Classify all the singular points (finite and infinite) of the following
Answer:
Writing the DE in standard form
are singular points. To classify what type of singularity, looking at
then
Hence, and
, therefore the singularity at
is
removable, hence
is a reqular singular point.
Now, looking at .
and
, therefore the
singularity at
is also removable, hence
is a reqular singular point.
To check the type of singularity, if any, at , the DE is first transformed using
| (1) |
This uses
| (2) |
and
Sustituting eqs (1,2,3) into the original DE gives
Writing the above in standard form
Expanding
Hence at there is a singularity (this means
). To find what type
And
Hence the singularity is removable. Therefore is a regular singular point.
problem: Classify all the singular points (finite and infinite) of the following
Answer:
Writing the DE in standard form
problem: Classify and
of the following
Answer:
problem: Classify and
of the following
Answer:
Problem: Estimate the number of terms in the Taylor series (3.2.1) and (3.2.2) that are necessary to compute
and
correct to three decimal places at
Answer:
Problem: How many terms in the Taylor series solution to with
are needed to evaluate
correct to three decimal
places?
Answer:
Problem:
Find series expansions of all the solutions to the following differential equations about .
Try
to sum in closed form any infinite series that appear.
Answer:
Derive (3.4.28). Where 2.4.28 is the solution of example 5 which is stated here:
Local behavior of solutions near an irregular singular point of a general nth-order Schrodinger equation. In this example we derive an extremely simple and important formula for the leading behavior of solutions to the nth-order Schrodinger equation
near an irregular singular point at
The exponential substitution and the asymptotic approximations
as
for
give the asymptotic differential equation
.
Thus,
, where
is an nth root of unity. This
result determines the
possible controlling factors of
. The leading behavior of
is
found in the usual way (see Prob. 3.27) to be
Answer:
Find the leading behaviors as of the following equations
Answer:
Find the leading behaviors as of the following equations
Answer:
Find the leading asymptotic behaviors as of the following equations
Answer:
What is the leading behavior of solutions to as
? Show that it is
inconsistent to assume that
However, show that the approximate equation
can be solved exactly by assuming a solution of the form
Answer:
Find the leading behavior as of the general solution to each of the following
equations
Answer: