Consider the following singular ODE $$ (t^{N-1}g(y'))'=t^{N-1}f(y) $$ with initial condition $$y(0)=y_0$$ $$y'(0)=0$$. where $g$ is and increasing function. How one can prove that this problem has a solution?

  • $\begingroup$ Why do you believe it has one? $\endgroup$ – Igor Rivin Oct 4 '11 at 14:26
  • $\begingroup$ It is difficult to understand both variables and functions, please, define them. In any case, it looks like homework. $\endgroup$ – mikitov Oct 4 '11 at 14:28
  • $\begingroup$ $g$ and $f$ are increasing function. $t$ is the variable and $y$ the unknown function. $\endgroup$ – Lugica Oct 4 '11 at 14:32
  • $\begingroup$ @mikitov's theory that this is homework is still not disproved. Voting to close until it is... $\endgroup$ – Igor Rivin Oct 4 '11 at 14:35
  • 1
    $\begingroup$ Ah, OK. You might get better responses if you add this explanation to your question. $\endgroup$ – Igor Rivin Oct 4 '11 at 15:15

One cannot prove existence in the generality that you have stated it. For example, if $g(p) = e^p$, there is no solution when $N>1$ and $f$ is differentiable.

To see this, observe that, if there were a solution $y(t)$ to your problem in a neighborhood of $t=0$, then the curve $(t,y,p) = (t,y(t),y'(t))$ would be an integral curve of the system $$ dy - p\ dt = d(t^{N-1}g(p)) - t^{N-1}f(y)\ dt = 0 $$ that passes through the point $(t,y,p) = (0,y_0,0)$.

Now, off the hypersurface $t=0$ in $typ$-space, this is equivalent to the system $$ dy - p\ dt = \bigl((N{-}1)g(p)-tf(y)\bigr)\ dt + tg'(p)\ dp = 0. $$ As long as $g(0)\not=0$ and $N>1$, this system has rank $2$ at the point $(t,y,p) = (0,y_0,0)$, so that there is a unique integral curve of the system passing through this point. However, inspection shows that the curve is $(t,y,p) = (0,y_0,p)$ is an integral curve that passes through the point in question. By uniqueness, it is the only one.

Thus, there is no integral curve of the kind that you would get from a solution to your problem.

I suspect that you have left out some hypotheses or not made a correct symmetry reduction of the system that you are trying to study.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.