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. 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. Given a homogeneous second order linear differential equation and two continuous linear independent solutions u(x) and v(x) with x0 and x1 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. Since u {displaystyle displaystyle u} and v {displaystyle displaystyle v} are linearly independent it follows that the Wronskian W [ u , v ] {displaystyle displaystyle W} must satisfy W [ u , v ] ( x ) ≡ W ( x ) ≠ 0 {displaystyle W(x)equiv W(x) eq 0} for all x {displaystyle displaystyle x} where the differential equation is defined, say I {displaystyle displaystyle I} . Without loss of generality, suppose that W ( x ) < 0   ∀   x ∈ I {displaystyle W(x)<0{mbox{ }}forall {mbox{ }}xin I} . Then So at x = x 0 {displaystyle displaystyle x=x_{0}} and either u ′ ( x 0 ) {displaystyle u'left(x_{0} ight)} and v ( x 0 ) {displaystyle vleft(x_{0} ight)} are both positive or both negative. Without loss of generality, suppose that they are both positive. Now, at x = x 1 {displaystyle displaystyle x=x_{1}} and since x = x 0 {displaystyle displaystyle x=x_{0}} and x = x 1 {displaystyle displaystyle x=x_{1}} are successive zeros of u ( x ) {displaystyle displaystyle u(x)} it causes u ′ ( x 1 ) < 0 {displaystyle u'left(x_{1} ight)<0} . Thus, to keep W ( x ) < 0 {displaystyle displaystyle W(x)<0} we must have v ( x 1 ) < 0 {displaystyle vleft(x_{1} ight)<0} . We see this by observing that if u ′ ( x ) > 0   ∀   x ∈ ( x 0 , x 1 ] {displaystyle displaystyle u'(x)>0{mbox{ }}forall {mbox{ }}xin left(x_{0},x_{1} ight]} then u ( x ) {displaystyle displaystyle u(x)} would be increasing (away from the x {displaystyle displaystyle x} -axis), which would never lead to a zero at x = x 1 {displaystyle displaystyle x=x_{1}} . So for a zero to occur at x = x 1 {displaystyle displaystyle x=x_{1}} at most u ′ ( x 1 ) = 0 {displaystyle u'left(x_{1} ight)=0} (i.e., u ′ ( x 1 ) ≤ 0 {displaystyle u'left(x_{1} ight)leq 0} and it turns out, by our result from the Wronskian that u ′ ( x 1 ) ≤ 0 {displaystyle u'left(x_{1} ight)leq 0} ). So somewhere in the interval ( x 0 , x 1 ) {displaystyle left(x_{0},x_{1} ight)} the sign of v ( x ) {displaystyle displaystyle v(x)} changed. By the Intermediate Value Theorem there exists x ∗ ∈ ( x 0 , x 1 ) {displaystyle x^{*}in left(x_{0},x_{1} ight)} such that v ( x ∗ ) = 0 {displaystyle vleft(x^{*} ight)=0} . On the other hand, there can be only one zero in ( x 0 , x 1 ) {displaystyle left(x_{0},x_{1} ight)} , because otherwise v would have two zeros and there would be no zeros of u in between, and it was just proved that this is impossible.