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Are these two statements of Sturm Comparison Theorem equivalent?

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Consider the two statements of Sturm Comparison Theorem that I found in two different books:

Let $y(x)$ and $z(x)$ be solutions of $$y''+p(x)y=0,qquad z''(x)+q(x)z=0$$ respectively and $p,q$ are continuous and $p(x)ge q(x),forall x$. Then $y$ vanishes at least once between any two successive zeros of $z$, unless $pequiv q$ and $y(x)=cz(x),forall x$.

---

Let $u(t),v(t)$ be nontrivial solutions of $$(p_1(t)u')'+q_1(t)u=0,qquad (p_2(t)v')'+q_2(t)v=0$$ respectively and let $p_1(t)ge p_2(t)>0$ and $q_2(t)ge q_1(t)$. Then between any two successive zeros $u(t)$, there exists at least one zero of $v(t)$, except when $u=cv$ for some $c$.

Are these two statements equivalent?

ordinary-differential-equations

share|cite|improve this question

asked Mar 16 at 20:41

xyzxyz

315

$endgroup$

add a comment | 

0

$begingroup$

Consider the two statements of Sturm Comparison Theorem that I found in two different books:

Let $y(x)$ and $z(x)$ be solutions of $$y''+p(x)y=0,qquad z''(x)+q(x)z=0$$ respectively and $p,q$ are continuous and $p(x)ge q(x),forall x$. Then $y$ vanishes at least once between any two successive zeros of $z$, unless $pequiv q$ and $y(x)=cz(x),forall x$.

---

Let $u(t),v(t)$ be nontrivial solutions of $$(p_1(t)u')'+q_1(t)u=0,qquad (p_2(t)v')'+q_2(t)v=0$$ respectively and let $p_1(t)ge p_2(t)>0$ and $q_2(t)ge q_1(t)$. Then between any two successive zeros $u(t)$, there exists at least one zero of $v(t)$, except when $u=cv$ for some $c$.

Are these two statements equivalent?

ordinary-differential-equations

share|cite|improve this question

asked Mar 16 at 20:41

xyzxyz

315

$endgroup$

add a comment | 

$begingroup$

Consider the two statements of Sturm Comparison Theorem that I found in two different books:

Let $y(x)$ and $z(x)$ be solutions of $$y''+p(x)y=0,qquad z''(x)+q(x)z=0$$ respectively and $p,q$ are continuous and $p(x)ge q(x),forall x$. Then $y$ vanishes at least once between any two successive zeros of $z$, unless $pequiv q$ and $y(x)=cz(x),forall x$.

---

Let $u(t),v(t)$ be nontrivial solutions of $$(p_1(t)u')'+q_1(t)u=0,qquad (p_2(t)v')'+q_2(t)v=0$$ respectively and let $p_1(t)ge p_2(t)>0$ and $q_2(t)ge q_1(t)$. Then between any two successive zeros $u(t)$, there exists at least one zero of $v(t)$, except when $u=cv$ for some $c$.

Are these two statements equivalent?

ordinary-differential-equations

share|cite|improve this question

asked Mar 16 at 20:41

xyzxyz

315

$endgroup$

share|cite|improve this question

asked Mar 16 at 20:41

xyzxyz

315

share|cite|improve this question

asked Mar 16 at 20:41

xyzxyz

315

asked Mar 16 at 20:41

xyzxyz

315

asked Mar 16 at 20:41

xyzxyz

315