Timeline for Parametric ODEs - when do there exist solutions independent of the parameter?
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
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| S Aug 12, 2017 at 19:36 | history | suggested | Ali Taghavi |
I add a tag
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| Aug 12, 2017 at 19:18 | review | Suggested edits | |||
| S Aug 12, 2017 at 19:36 | |||||
| Aug 7, 2017 at 21:43 | answer | added | Robert Bryant | timeline score: 5 | |
| Aug 7, 2017 at 0:43 | comment | added | Igor Khavkine | A complementary suggestion to that of Michael Renardy is to consider the simultaneous system consisting of $P_{\lambda_i}=0$, where $\lambda_1$ through $\lambda_4$ are independent values of the parameter, and then eliminate $y$ through $y'''$ from this larger system. The resulting constraints on the coefficients should then be valid for any value of $\lambda_1$ through $\lambda_4$. | |
| Aug 7, 2017 at 0:35 | comment | added | Michael Renardy | In principle, you could consider P and its derivatives with respect to $\lambda$ of orders 1 through 4, then eliminate $y$, $y'$, $y''$ and $y'''$ from this system. The result would be an equation relating the coefficients which is a necessary condition for what you want. I doubt, however, that this is doable in practice. It would probably defeat the capabilities of symbolic manipulation software. | |
| Aug 6, 2017 at 21:27 | comment | added | Jeanne Clelland | No, I don't know y(x). I want to know conditions on the coefficients that would guarantee the existence of some solution y(x), but I don't have any restrictions on the form of the solution. | |
| Aug 6, 2017 at 21:01 | comment | added | Robert Bryant | Do you know $y(x)$ explicitly? Look at the vector space of polynomials of degree $5$ vanishing on the image curve $(y(x),y'(x),y''(x),y'''(x))$ in $\mathbb{R}^4$, say, $C$; its dimension is between $1$ (since you are supposing that at least one such polynomial exists) and $84-6=78$ (since $y(x)$ is not constant). There may be restrictions on the dimension since $C$ must be an integral curve of an Engel system (with some degeneracy locus) on $\mathbb{R}^4$. If the ideal of polynomials that vanish on $C$ is not principal, i.e., has independent generators, then $C$ is algebraic, which may help. | |
| Aug 6, 2017 at 19:53 | history | asked | Jeanne Clelland | CC BY-SA 3.0 |