Real Analysis 597564.Consider the change of variablex=cost. Then, by the chain rule,
dy
dx=
dy
dt
dx
dt=−
dy
dt
sintand
d^2 y
dx^2=
d^2 y
dt^2−
dy
dxd^2 x( dt^2
dx
dt) 2 =
d^2 y
dt^2
sin^2 t−
costdy
dt
sin^3 t.
Substituting in the original equation, we obtain the much simpler
d^2 y
dt^2+n^2 y= 0.This has the functiony(t)=cosntas a solution. Hence the original equation admits the
solutiony(x)=cos(narccosx), which is thenth Chebyshev polynomial.
565.We interpret the differential equation as being posed for a functionyofx. In this
perspective, we need to writed
(^2) x
dy^2 in terms of the derivatives ofy with respect tox.
We have
dx
dy
=
1
dy
dx,
and using this fact and the chain rule yields
d^2 x
dy^2=
d
dy⎛
⎜
⎝
1
dy
dx⎞
⎟
⎠=
d
dx⎛
⎜
⎝
1
dy
dx⎞
⎟
⎠·
dx
dy=−
1
(
dy
dx) 2 ·
d^2 y
dx^2·
dx
dy=−
1
(
dy
dx) 3 ·
d^2 y
dx^2.
The equation from the statement takes the form
d^2 y
dx^2⎛
⎜⎜
⎜
⎝
1 −
1
(
dy
dx