Undergrad Solution of Quantum differential equation

[Edge5]

(I think I couldn't add the image)
you can see my answer in link

https://pasteboard.co/HPKZ6KD.jpg

(Please first see my answer in the link)
But in answer it is φ= Asin(kx) + Bcos(kx)

I know that euler formula is e^ix = cosx +isinx

But I can't get this answer can you help me?

 

[Delta2]

In the answer key does it say if the constants A,B are real or complex?

My opinion is that if $\phi$ is complex valued then the general solution is as your answer says (and in your answer the constants A,B can be complex constants).
However if $\phi$ is real valued then the correct answer is as the answer key says that is $\phi=A\sin(kx)+B\cos(kx)$ where A,B are real constants here.

 

[Edge5]

In the answer key does it say if the constants A,B are real or complex?

My opinion is that if $\phi$ is complex valued then the general solution is as your answer says (and in your answer the constants A,B can be complex constants).
However if $\phi$ is real valued then the correct answer is as the answer key says that is $\phi=A\sin(kx)+B\cos(kx)$ where A,B are real constants here.

Assuming answer is real. How do I get from this general answer to $\phi=A\sin(kx)+B\cos(kx)$

 

[Delta2]

if we assume $\phi$ is real valued then from your general answer (for which i ll use $A'$ and $B'$ to denote the complex constants as well as $\phi'$ for the complex valued function) we ll have (i use Euler's formula to rewrite your general answer).

$$\phi'=A'\cos(kx)+B'\cos(-kx)+i(A'\sin(kx)+B'\sin(-kx))=(A'+B')\cos(kx)+i(A'-B')\sin(kx) \text{(1)}$$

So we just looking for complex constants $A',B'$ such that $(A'+B')=B (2)$ and $(A'-B')=-iA (3)$ and then for these constants it would be $\phi=\phi'$. The system of equations (2),(3) has unique solutions for $A',B'$ given that $A,B$ are real.

 

[Delta2]

Now that i think of it again, if we allowed for $\phi$ to be complex valued (since it is a wave function it would be complex valued in general case), and also allow for constants A,B to be complex in the answer key (the answer that your book says), then the answer key and your answer are equivalent.