1.The ODE X"+ kX = 0 has different types solution depending on sign of k.
We consider the three possible cases separately.
k=0: X"=0 so that X(x)=Ao+Box, X'(0)=0 implies Bo=0 so that X(x)=Ao. Finally, X(1)=0 implies Ao=0 and there is only trivial solution X=0.
The matrix has determinant e^mu + e^-mu= 2cosh.mu, which is never zero and it follows that pair of equations has unique solution Ahat=Bhat=0 and only trivial solution X=0.
A differential equation and a matrix determinant are investigated.