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## Homework Statement

1. Let D and I be the differential operator and the identity operator, respectively. Find two

real-valued functions f(t) and g(t) such that:

D[tex]^{2}[/tex] + I = (D + f(t)I)(D + g(t)I):

2. Use this theorem to prove the corollary given below.

Theorem: There are two nonzero solutions y[tex]_{1}[/tex] and y[tex]_{2}[/tex] to the differential equation

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

such that one of the two functions is not a constant multiple of the other, and

that c[tex]_{2}[/tex]y[tex]_{1}[/tex]+c[tex]_{2}[/tex]y[tex]_{2}[/tex] for arbitrary constants c[tex]_{1}[/tex] and c[tex]_{2}[/tex] is a general solution to the

differential equation.

Corollary: If z[tex]_{1}[/tex] and z[tex]_{2}[/tex] are two nonzero solutions to the differential equation such that one of the two functions is not a constant multiple of the other, then c[tex]_{1}[/tex]z[tex]_{1}[/tex] + c[tex]_{2}[/tex]z[tex]_{2}[/tex] for arbitrary constants c[tex]_{1}[/tex] and c[tex]_{2}[/tex] is a general solution the

differential equation.

3. Prove the theorem stated in #2.

## Homework Equations

1. I know there is a theorem such that ((D[tex]^{2}[/tex] + a[tex]^{2}[/tex]I) = 0 and then this equation equals c[tex]_{1}[/tex]cos(at) + c[tex]_{2}[/tex]sin(at)

## The Attempt at a Solution

1. I tried to do this, but got lost. I know that

(D[tex]^{2}[/tex] + I) = c[tex]_{1}[/tex]cos(1t) + c[tex]_{2}[/tex]sin(1t).

That's all I've got.

2. My teacher assigned this problem, but I have no clue how to go about proving this corollary. The corollary seems like the same as the theorem except with using z's instead of y's and I'm confused.

3. Again, I have no clue how to prove this theorem, so I need some help.

Thanks guys!