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My DE course uses an online homework service to distribute and collect homework. One of the problems in this set is to furnish an autonomous DE consistent with the phase portrait below:

enter image description here

I came up with $y' = (y-1)^2(y-3)^2(y-5)^2$, which the program rejected. Have I made a mistake, or has the program?

Edit: Another problem was to find an autonomous DE consistent with the phase portrait

enter image description here

For this, I came up with $(y-1)(y-3)^2(y-5)$. I expanded this into $y^4 - 12y^3 + 50y^2 - 84y + 45$ and got marked correct. When I expanded $(y-1)^2(y-3)^2(y-5)^2$ into $225-690 y +799 y^2 - 444 y^3+127$ $ y^4 -18 y^5 +y^6$ using Walpha, I got marked wrong.

Is there some error I'm making? (And do problems like this actually arise in "real life"?!)

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Are the tickmarks at $1$, $3$ and $5$ supposed to imply that those are equilibrium points, or are they just the scale on the axis? – Robert Israel Feb 3 '13 at 3:24
@RobertIsrael I was assuming they were the zeroes of $f(y)$ in $y' = f(y)$. (Maybe that's it?) – Chris Feb 3 '13 at 3:25
@Robert-israel:these are clearly equilibrium points but how find out these are sink points or source points – Maisam Hedyelloo Feb 3 '13 at 3:35
What sort of information does the program require as input? In particular, are you supposed to set any information about the points 1,3,5, and if so, what options are you allowed to tick off for them? Is there any particular reason the case $y>5$ is cut off? – anon Feb 3 '13 at 4:12
@anon None whatsoever. (I'll spare you a screen cap....) – Chris Feb 3 '13 at 4:13

Apparently, the error in the first case was in thinking that $5$ should be a critical point; in fact, it's just a tickmark. If the tickmarks are not required to be critical points, $y'=1$ should be as acceptable as $y'=(y-1)^2(y-3)^2$.

In the second case, $y'$ has to change sign around $y=1$ and $y=5$. The ODE $y'=(y-1)(y-5)$ matches the sign pattern. So does $y'=(y-1)(y-5)(y-3)^2$, but again, having a critical point at $3$ should not be a requirement.

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