- 321

- 20

**1. The problem statement, all variables and given/known data**

Find the general solution of the second order DE.

[itex] y'' + 9y = 0 [/itex]

**2. Relevant equations**

**3. The attempt at a solution**

Problem is straight forward I just don't get why my answer is different than the books.

So you get

[itex] m^2 + 9 = 0 [/itex]

[itex] m = 3i [/itex] and [itex] m = -3i [/itex]

so the general solution would be:

[itex] c_1e^{3ix} + c_2e^{-3ix} = y [/itex]

my book gives me

[itex] e^{i\theta} = cos(\theta) + isin(\theta) [/itex]

from there I get

[itex] e^{iβx} = cos(βx) + isin(βx) [/itex]

[itex] e^{i-βx} = cos(βx) - isin(βx) [/itex]

I have

[itex] e^{i3x} = cos(3x) + isin(3x) [/itex]

[itex] e^{i-3x} = cos(3x) - isin(3x) [/itex]

so I get [itex] y = c_1cos(3x) + c_1isin(3x) + c_2cos(3x) -c_2isin(3x) [/itex]

but my book gives me

[itex] y = c_1cos(3x) + c_2sin(3x) [/itex]

I feel like my answer is still valid for some reason.. I just don't know how they got their answer from my answer. I used the correct identity..