(PDE)(ODE)(Sturm-Liouville) I'm ending up having to use Cauchy-Euler in solving an SLDE, and upon reviewing the technique, I fail to understand how the general solution is arrived at without imaginary numbers. More info below.

Question

mendicant_bias · Answer

I understand deriving the solution for Cauchy -Euler problems up until the point where $$y=\cos(\ln(\alpha)x)+i \sin(\ln(\alpha)x)$$

mendicant_bias · Answer

What I don't understand is how the homogeneous solution y_h is achieved with no imaginary i attached to the sine term, e.g.

mendicant_bias · Answer

(Now I'm getting more confused, lol, gimme a minute)

mendicant_bias · Answer

Getting from here: http://tutorial.math.lamar.edu/Classes/DE/EulerEquations_files/eq0025M.gif to here: http://tutorial.math.lamar.edu/Classes/DE/EulerEquations_files/eq0026M.gif

freckles · Answer

I think c1 and c2 can be complex numbers

mendicant_bias · Answer

@SithsAndGiggles

anonymous · Answer

I'm @freckles on this one, the constants are not restricted to the reals.

mendicant_bias · Answer

Let me post what my book says, because I dunno, that just sounds odd: http://i.imgur.com/4XdQn8g.png

mendicant_bias · Answer

"We wish to write the solution in terms of real functions only"

mendicant_bias · Answer

brb real quick

mendicant_bias · Answer

Back

mendicant_bias · Answer

But yeah, what about what it says? You're no doubt better at interpreting math than me, but I was strongly under the impression that the constants could not be imaginary in any way. @SithsAndGiggles

freckles · Answer

http://tutorial.math.lamar.edu/Classes/DE/ComplexRoots.aspx read the thing before the examples it says c1 and c2 can be complex