Hi the first question should be this.
i) Do we say that the equation given has no solution because there are two different values of c2?
No. There is no solution, because you have shown that C_2 must be equal to two different values at the same time, which is a contradiction.
Why do you need another way to verify there is no solution? Have you not convinced yourself? (What I'm really saying, of course, is that I don't know of another way to verify it. I suppose you might look into the instability that a shooting numerical method would probably give you.)