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

Asked by Yingquan Li

Yingquan Li (view profile)

on 25 Sep 2011
Latest activity Commented on by darova

darova (view profile)

on 14 Sep 2019
Accepted Answer by John D'Errico

John D'Errico (view profile)

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.

Answer by John D'Errico

John D'Errico (view profile)

on 13 Sep 2019

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.

Walter Roberson

Walter Roberson (view profile)

on 13 Sep 2019
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
John D'Errico

John D'Errico (view profile)

on 13 Sep 2019
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.
darova

darova (view profile)

on 14 Sep 2019
Finally! 8 years, guys
Congratulation

Answer by Walter Roberson

Walter Roberson (view profile)

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.