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I'm looking for an explicit (not numeric) solution to the following non-linear delay differential equation (aka difference differential equation). It's a sort of Riccati type equation.

$\frac{y_x(x)}{y(x)} = y(x-d) - y(x) - \rho y(x)$,

where $d>0$ is a constant parameter for the delay and $\rho>0$ a positive constant parameter. I'm also interested in $d<0$, actually.

First, note that the division by $y(x)$ on the left-hand side makes the equation non-linear. Secondly, note that the special case $d=0$ is an ordinary differential equation of the Riccati type, which can be solved. Thirdly, for any $d \neq 0$, I have solved the special case $\rho=0$ (It's probably not much of a feat since I managed it). The way I found the solution is rather roundabout, relying on symmetries. The solution is a ratio of sums of exponential functions. It would be a total pain to copy the solution method here, so I hope it won't be necessary ;-)

And my question is: can anyone see a way to handle the general case $\rho>0$? Or a special case of it, like $\rho=1$ for instance.

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If anyone knows a reference to special solutions of Riccati differential equations with a delay term, that would be great. Special cases of Riccati can clearly be solved (the case $\rho=0$ for instance) Seeing other solved examples might inspire an approach... –  PatrickT Apr 25 '12 at 12:54
    
Also, this is my first post on math.stackexchange, so if you think I didn't ask the question in the right way, please do let me know, I'm happy to learn. If you think I should ask this sort of question elsewhere, so too please do let me know. Thanks. –  PatrickT Apr 25 '12 at 12:56

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