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A particular differential equation... why can't I solve it?

Asked by Yingquan Li on 25 Sep 2011
Latest activity Commented on by darova
on 14 Sep 2019 at 13:26
the differential equation: dy/dt = (t-e^-t)/(y+e^y) was assigned by my teacher and supposedly dsolve() can solve it, resulting in an implicit solution. No matter what I try, I think the teacher is wrong because I just get the empty matrix, which according to the documentations means that no solution could be found. Any thoughts? I'm getting nowhere digging in the documentation.

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2 Answers

Answer by John D'Errico
on 13 Sep 2019 at 21:27
 Accepted Answer

Easier than you might think to solve, even with pencil and paper. But sometimes a computer won't see the trick, at least, not without help. I've seen cases where that happens, but not here. Of course, since this is now an 8 year old, unanswered question. it may also be that dsolve has become smarter since it was originally posed too.
Here, I think it is possible the transcription error was Walter's fault in what he tried, because dsolve succeeds.
syms t y(t)
>> dsolve(diff(y(t), t) == (t-exp(-t))/(y(t)+exp(y(t))))
Warning: Unable to find explicit solution. Returning implicit solution instead.
> In dsolve (line 208)
ans =
solve(2*exp(y) + y^2 == 2*C8 + 2*exp(-t) + t^2, y)
So the solution is indeed an implicit euation. How would we arrive at it without the help of MATLAB? This is a separable equation, if you multiply by the denominators (y + exp(y))*dt. So we have the problem...
(y + exp(y)) dy = (t - exp(-t)) dt
Integrating each side, we get
y^2 / 2 + exp(y) = t^2 /2 + exp(-t) + C
If you now multiply by 2, you should see it is the same implicit problem returned by dsolve. C is of course an unknown constant of integration.

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Ah, I see I had a t at a place I should have had y(t)
Maple returns
t^2/2 + exp(-t) - y(t)^2/2 - exp(y(t)) + _C1 = 0
Yes. I thought it may have been a transcription error indeed. Its an easy mistake to make too, because the eye sees the exponential, and one mentally puts in a t there to match the numerator.
Finally! 8 years, guys
Congratulation

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Answer by Walter Roberson
on 25 Sep 2011

In Maple,
dsolve(diff(y(t), t) = (t-exp(-t))/(y(t)+exp(t)));
returns empty as well. That suggests that perhaps the equation is not transcribed correctly.

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