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We want to find functions

p(t)

q(t)

f(t)

such that
(a)

p

and

q

are continuous functions on the open interval

(0,\pi)

. (b)

f

is an infinitely differentiable nonzero function on the whole real line

(-\infty,\infty)

such that

f(0)=f'(0)=f''(0)

. (c)

y=\sin t

and

y=f(t)

are solutions of the differential equation

y''+p(t)y'+q(t)y=0

(0,\pi)

.
Is this possible? Either prove this is not possible, or show this is possible by providing an explicit example of such

f,p,q

Problem Statement