Sturm separation theorem
In mathematics, in the field of ordinary differential equations, Sturm separation theorem, named after Jacques Charles François Sturm, describes the location of roots of solutions of  homogeneous second order linear differential equations. Basically the theorem states that given two linear independent solutions of such an equation the zeros of the two solutions are alternating. Sturm separation theorem 

If u(x) and v(x) are two non-trivial continuous linearly independent solutions to a homogeneous second order linear differential equation with x0 and x1 being successive roots of u(x), then v(x) has exactly one root in the open interval (x0, x1). It is a special case of the Sturm-Picone comparison theorem. Proof 

Since  and  are linearly independent it follows that the Wronskian  must satisfy  for all  where the differential equation is defined, say . Without loss of generality, suppose that . Then

So at 

and either  and  are both positive or both negative. Without loss of generality, suppose that they are both positive. Now, at  

and since  and  are successive zeros of  it causes . Thus, to keep  we must have . We see this by observing that if  then  would be increasing (away from the -axis), which would never lead to a zero at . So for a zero to occur at  at most  (i.e.,  and it turns out, by our result from the Wronskian that ). So somewhere in the interval  the sign of  changed. By the Intermediate Value Theorem there exists  such that . On the other hand, there can be only one zero in , because otherwise  would have two zeros and there would be no zeros of  in between, and it was just proved that this is impossible. References 

Ordinary differential equations
Theorems in analysis