I have a question relating to the Euler equation,

X^2Y'' + AXY' +BY = 0.

I understand that there are three possible solutions for Y;
Y=c1x^m1 + c2x^m2 for distinct roots
Y=[c1 + c2ln(x)]x^m for repeated roots
or Y=[c1cos(bln(x))+c2sin(bln(x))]x^a for complex conjugate roots

I am trying to show that there are no solutions (real or imaginary), other than Y=0 to an euler Equation which satisfies the boundary conditions Y(1) = 0, Y(2) = 0.

When I put the boundary conditions Y(1) = 0, Y(2) = 0 into each of the possible solutions of Y, the only solution I can get is Y=0.

Are there any other possible solutions to an euler-cauchy problem than the three I mentioned above that should be considered?

I have noticed in every boundary value euler cauchy problem has the boundary conditions with Y and the derivative of Y, rather than Y in both. Could this mean there are no solutions?

I hope this makes sence, I am a little confused by this problem.