Exercise 2.7.2

Answers

1.

No. Let

W

be a finite-dimensional subspace generated by the function

y = t

. Thus

y

is a solution to the trivial equation

0 y = 0

. But the solution space is

C^{\infty}

but not

W

. Since

y^{(k)} = 0

for

k \geq 2

and it is impossible that

a y^{'} + by = a + bt = 0

for nonzero $a$ , $W$ cannot be the solution space of a homogeneous linear differential equation with constant coefficients.

2.

No. By the previous argument, the solution subspace containing

y = t

must be

C^{\infty}

3.

Yes. If

x

is a solution to the homogeneous linear differential equation with constant coefficients whose is auxiliary polynomial

p (t)

, then we can compute that

p (D) (x^{'}) = D (p (D) (x)) = 0

4.

Yes. Compute that

p (D) q (D) (x + y) = q (D) p (D) (x) + p (D) q (D) (x) = 0 .

5.

No. For example,

e^{t}

is a solution for

y^{'} - y = 0

and

e^{- t}

is a solution for

y^{'} + y = 0

, but

1 = e^{t} e^{- t}

is not a solution for

y^{″} - y = 0

jephianlin

2011-06-27 00:00