I have some doubts about this kind of second - order differential equation, which is used a lot in physics and for which there are many topics (but in this case the situation is a bit different because k is in general complex number):
I have many doubts about the solutions of this equation, because I have seen different expressions for them in several cases (all applied to electromagnetic problems):
1)I have seen this kind of solution (it is the expression used to describe voltage or current along a transmission line):
where T1 and T2 are complex values.
So, from this kind of analysis, I'll say that:
The solution is a complex linear combination of exponential functions with arguments kx and -kx, with k complex (because in general we have supposed k complex from the beginning).
So, my first question is: is this true for any case? Or may the solution be different depending on k?
2) In other situations (for instance analysis of rectangular and circular waveguides) I have seen different solutions:
with T0 complex value.
Second question: is this solution equivalent to that seen in 1)? And is it true for any value of k?
3) I have seen also another kind of solution:
So my third question is: is this solution equivalent to that seen in 1) and 2)? And is it true for any value of k?
Then, I have a last question: I have seen that these solutions have been used in different situations, by specifing the domain of x. If x was defined in a bounded domain, I have usually seen solutions shown in 1), while for unbounded domains, I have usually seen 3). I wanted to know if it is just a reason of convenience for those specific applications, or if it is a strict math rule.
First note that
$$sinh(kx) = \frac{e^{kx}-e^{-kx}}{2i}$$ and $$cosh(kx) = \frac{e^{kx}+e^{-kx}}{2}$$
this is a defintion of these functions.
So we can see that these definitions nearly look like
$$ T_1 \cdot e^{kx} + T_2 \cdot e^{-kx}$$
we just have to choose $T_1$ and $T_2$ so that we get multiples of $cosh(kx)$ and $sinh(kx)$, getting your third solution from the first. I'll leave that to you to try and figure out :)
Your second solution looks like it might be slightly off(I may be wrong and hopefully someone can correct me if I am, but I'll work through it).
We note that
$sinh(kx) = -isin(ikx)$ and $cosh(kx) = cos(ikx)$,
this just comes from the definitions of the Hyperbolic and Trigonometric functions.
So your third solution becomes;
$$T_2 \cdot cos(ikx) - T_1 \cdot i \cdot sin(ikx)$$
Now all we need to do is apply what is called a Harmonic Identity, this one in particular says that,
$$Acos(x) - Bsin(x) = Rcos(x+\alpha)$$
where $R = \sqrt{A^2 + B^2}$ and $\alpha = arctan(\frac{b}{a})$.
So now you apply that identity to what we have above with $A = T_2$, $B = T_1 \cdot i$ and $x = ikx$ and your second answer should pop out!
If what I've done is right it should be of the form $T_0cos(ikx+\alpha)$ instead.
I hope this helps :)