- #1
QuarkCharmer
- 1,051
- 3
Homework Statement
[tex]3\frac{d^{2}y}{dx^{2}} + 2\frac{dy}{dx} + y = 0[/tex]
Homework Equations
The Attempt at a Solution
[tex]3y'' + 2y' +y = 0[/tex]
I know the solution is going to be in the form of [tex]y=Ce^{mx}+De^{nx}+...[/tex]
(Unless there is a multiplicity, in which case I understand that too)
So I'll just skip to the Aux Equation:
[tex]3m^{2} + 2m + 1 = 0[/tex]
[tex]m = \frac{-1+\sqrt{2}i}{3} or \frac{-1-\sqrt{2}i}{3}[/tex]
Thus, a general form of the solution is:
[tex]y(x) = C_{1}e^{\frac{-1}{3}}e^{\frac{\sqrt{2}ix}{3}} + C_{2}e^{\frac{-1}{3}}e^{\frac{-\sqrt{2}ix}{3}}[/tex]
So now I just rename the constants..
[tex]Ae^{\frac{\sqrt{2}ix}{3}} + Be^{\frac{-\sqrt{2}ix}{3}}[/tex]
and since...
[tex]e^{ix} = cos(x) + isin(x)[/tex]
[tex]A(cos(\frac{\sqrt{2}x}{3})+isin(\frac{\sqrt{2}x}{3})) + B(cos(\frac{-\sqrt{2}x}{3})+isin(\frac{-\sqrt{2}x}{3}))[/tex]
Since cosine is even and sine is odd...
[tex]Acos(\frac{\sqrt{2}x}{3}) + iAsin(\frac{\sqrt{2}x}{3}) + Bcos(\frac{\sqrt{2}x}{3}) -iBsin(\frac{\sqrt{2}x}{3})[/tex]
Usually here is where the imaginary stuff cancels out, but it's just not happening this time and I think I am making a mistake somewhere but I can't see it. What do I do??