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I have a complicated 3rd-order ODE of the form $P(y, y', y'', y''') = 0$, where $P$ is a complicated polynomial (5th-order with 24 terms) and coefficients that are (unknown) functions of a parameter $\lambda$, say $$ P(z_0, z_1, z_2, z_3) = \sum c_{j_0 j_1 j_2 j_3}(\lambda) z_0^{j_0} z_1^{j_1} z_2^{j_2} z_3^{j_3}. $$ What I want to know is: Suppose that there exists a non-constant solution $y(x)$ that is independent of $\lambda$. What conditions does this force the coefficient functions $c_{j_0 j_1 j_2 j_3}(\lambda)$ to satisfy?

(An easy analog would be something like: a 1st-order ODE of the form $$y' + c_0(\lambda) y = 0 $$ has nonconstant parameter-independent solutions if and only if $c_0(\lambda)$ is a constant function.)

I'm sure that the specific conditions I'm looking for depend on the precise form of the ODE, which is pretty daunting in this case. I'm just wondering if there's a reasonable algorithm I could apply to find them.

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  • $\begingroup$ 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. $\endgroup$ Aug 6 '17 at 21:01
  • $\begingroup$ 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. $\endgroup$ Aug 6 '17 at 21:27
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    $\begingroup$ 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. $\endgroup$ Aug 7 '17 at 0:35
  • $\begingroup$ 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 $\lambda_4$. $\endgroup$ Aug 7 '17 at 0:43
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Depending on what you know about the coefficients $c_{j_0j_1j_2j_3}(\lambda)$, I think that it's not as hopeless as all that.

First of all, such a curve would have to lie in the common zero locus $Z\subset\mathbb{C}^4$ of the polynomials $P_\lambda(z)$. This locus is the same as the space of common zeroes of all the polynomials in the linear span $L$ of the $P_\lambda(z)$ in the space of quintic polynomials. Thus, consider the ideal $I\subset\mathbb{C}[z_0,z_1,z_2,z_3]$ generated by this linear span of quintics. Assuming that you can compute this (i.e., find a basis for it), there are fast algorithms (using Gröbner bases and Macaulay for example) for determining the dimension $Z$.

Case 0: If $Z$ is empty or its dimension is zero, then there is no nonconstant solution $y(x)$ whose graph $\bigl(y(x),y'(x),y''(x),y'''(x)\bigr)$ lies in $Z$.

Otherwise, decompose $Z$ into its irreducible components (again, Gröbner and Macaulay can be very helpful here) and treat each component of $Z$ separately. From now on, I'll assume that $Z$ is irreducible and is defined by a reduced ideal (i.e., you have found a basis for the ideal of polynomials that vanish on $Z$).

Suppose that the dimension of $Z$ is at least $1$, and consider the Pfaffian system $I$ generated by the $1$-forms $$ \zeta_0 = z_2\,\mathrm{d}z_0 - z_1\,\mathrm{d}z_1\,, \quad\text{and}\quad \zeta_1 = z_3\,\mathrm{d}z_0 - z_1\,\mathrm{d}z_2\,, \quad\text{and}\quad \zeta_2 = z_3\,\mathrm{d}z_1 - z_2\,\mathrm{d}z_2\,. $$ This system will have rank $2$ everywhere except along the locus $z_1=z_2=z_3=0$, which you don't care about anyway, since this would correspond only to $y(x)$ being constant. In fact, you only care about the part of $Z$ that is not contained in the hyperplane $z_1=0$, so I'll assume from now on that we have removed this hyperplane. Nearly all of the integral curves of $I$ that have $z_1$ not identically vanishing are locally graphs of the form $\bigl(y(x),y'(x),y''(x),y'''(x)\bigr)$, and you can easily characterize the exceptions (such as, for example, $z_0$ and $z_1$ and $z_2$ are constant), so you can incorporate the test to throw those out into your algorithm, so that you get only the so-called 'admissable curves'.

Case 1: If the dimension of $Z$ is $1$ and it is not an integral curve of $I$ (i.e., the $1$-forms $\zeta_j$ are not in the differential ideal generated by the polynomials that vanish on $Z$), then there is no admissable curve in $Z$.

Case 2: If the dimension of $Z$ is $2$, then you need to look at the locus $Z'\subset Z$ on which either $Z$ is singular or on which $I$ pulls back to have rank at most $1$. If $Z'$ has dimension $2$, then $Z'$ is foliated by integral curves of $I$, and the admissable ones will be the $3$-jet graphs of the curves you seek. If $Z'$ has dimension $1$, then repeat Case 1 with $Z'$ in the place of $Z$.

Case 3: If the dimension of $Z$ is $3$, then the smooth part of $Z$ is foliated by integral curves of $I$, and each admissable integral will correspond to a $y(x)$ that satisfies all the $P_\lambda(z)$. The singular locus of $Z$ will have dimension at most $2$, so, for that, you are reduced to Case 2 or Case 1 (or Case 0).

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  • $\begingroup$ Robert, thanks so much! I'll have to think about this. The big problem is that I know almost nothing about the functions $c_{j_0 j_1 j_2 j_3}(\lambda)$, because they themselves are expressed in terms of three unknown functions of $\lambda$ and their derivatives up to order 2. Ultimately I want to know necessary conditions on these 3 unknown functions of $\lambda$ for the zero locus $Z$ you describe above to have dimension at least 1. $\endgroup$ Aug 8 '17 at 1:29

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