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These are questions asked by Steven_Huang

I am wondering if Maple DETools package has functions or command to deal with the following problem: algebraic invariant curve. Some first order ODE preserves such type of curve as their solutions. For example, the following ODE has an algebraic curve y(x)=0 as its particular solutions:

> odetest(y(x)=0, y'(x)=y(x)^3-2*x*y(x)^2,y(x));

> 0

The ODE in general does not have algebraic solutions. The solutions are computed in terms of special functions. In some cases the algebraic curve could have multi-variate forms . I am wondering about one question: Does Maple have tools to find solutions of algebraic curve for ODE, without knowing the information of general solutions? I have already tried PDETools:-casesplit, but it seems to classify such curves to the same case to the general solution. 

I will be glad if anyone could give me some advice.

I was trying to solve a system of eight cubic equations, with eight variables. Note that the particular solutions should exist and my goal is to find all  possible solutions using solve. However, when executing, the solve keeps running for a whole day and did not throw any results. I also set the parameter infolevel[solve] to be 3 and find out that it is stuck in the step "GroebnerBasis: computing a factored plex basis using Groebner[Solve]". Can anyone tell me how to deal with that? Here's the Maple file.

I was trying to solve a system of polynomial equations, which contains three equations and six variables $a_0,a_1,a_2,b_0,b_1,b_2$. However, as I swap the variable name, Maple solve function gives me a totally different solutions. Only the solutions before swapping the variables are useful for the problem I study. I have already attached the file. Could anyone tell me if the choice of variable name really matters, or if i just misuse this function?

polysols(diff(u(x), x) = u(x)^2 - 1) produces no results, while it can be verified by direct observation that u(x) = 1 is a polynomial solution.

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