3.4E: Exercises
This page is a draft and is under active development.
( \newcommand{\kernel}{\mathrm{null}\,}\)
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Exercise
(a) Adapt the proof of Theorem
on
on
(b) Use part (a) to show that the general solution of
Exercise
Use reduction of order to show that if
has a repeated root
on
- Answer
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If
, then (A) .
If , then
and , so
from (A). Therefore,\) u''=0\) , so \) u=c_1+c_2x\) and
.
Exercise
A nontrivial solution of
is said to be
has oscillatory solutions on
Exercise
In Example
(a) Introduce the new variables
which has a regular singular point at
(b) Introduce the new variables
which has a regular singular point at
- Answer
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a. If
and , then
,
so satisfies Legendre's equation if and only if satisfies
(A) \) \displaystyle t(2+t){d^2Y\over dt^2}+2(1+t){dY\over
dt}-\alpha(\alpha+1)Y=0\) . Since (A) can be rewritten as
\) \displaystyle t^2(2+t){d^2Y\over dt^2}+2t(1+t){dY\over
dt}-\alpha(\alpha+1)tY=0\) , (A) has a regular singular point at
\) t=0_0\) .
b. If and , then
,
so satisfies Legendre's equation if and only if satisfies
(B) \) \displaystyle t(2-t){d^2Y\over dt^2}+2(1-t){dY\over
dt}+\alpha(\alpha+1)Y\) ,
Since (B) can be rewritten as
(B) ,
(B) has a regular singular point at
.
Exercise
Let
Let
(a) Show that
where
(b) Show that