- #1

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## Homework Statement

Let

[tex]V=span(sinx,cosx)[/tex]

be the subspace of Maps(R,R) generated by the functions sin(x) and cos(x), and let

[tex]D:V \to V[/tex]

be the differential operator defined by

[tex]D(y)=y''+y'+y[/tex] for [tex]y E V[/tex].

Show that [tex]Im(D) = V[/tex] and conclude that for every [tex]f E V[/tex], the differential equation

[tex]f=y''+y'+y[/tex]

has a solution [tex]y E V[/tex].

## Homework Equations

Not really any, you need Euler's formula to solve the DE though.

## The Attempt at a Solution

The differential equation doesn't make any sense to me in that form, so after some research into solving such things (I have never seen one before), I solved

[tex]0=y''+y'+y[/tex]

and obtained

[tex]y(f)= c_1e^{\frac{-f}{2}}sin(\frac{\sqrt{3}}{2}f)+ c_2e^{\frac{-f}{2}}cos(\frac{\sqrt{3}}{2}f)[/tex]

Which is pretty cool, but I'm not entirely sure if that helps me at all. I mean, it looks like a linear combination of things, which is good maybe. Also, can you just make it equal to zero like that?