# DSolve does not solve a nonlinear differential equation [closed]

I am trying to solve the following nonlinear differential equation. Mathematica does not give any solution.

uap = uat*Exp[-I*w*(t - X*Cos[theta]/c)]
pde =
-I*w*Ze[X, t] + (ua*Cos[theta] + uap*Cos[theta])* D[Ze[X, t], X] +
Sl*D[Ze[X, t], X]^2 - uap*Sin[theta]
DSolve[pde == 0, Ze[X, t], {X, t}]


The output is the same as input statement DSolve with pde substituted from the above variables, with a warning

Solve::ifun: Inverse functions are being used by Solve, so some solutions may not be found; use Reduce for complete solution information. >>

Can anyone please suggest what is wrong with the input command. I am new to mathematica and I followed instructions in the documentation. I was able to reproduce the examples given in documentation, but this particular problem doesn't work.

• Try DSolve (capital S) – Ulrich Neumann Feb 5 '18 at 11:38
• @ Ulrich Thank you for pointing out the typo. I used DSolve[ ] in my mathematica notebook. Mathematica doesn't give any solution but a warning Solve::ifun: Inverse functions are being used by Solve, so some solutions may not be found; use Reduce for complete solution information. >> and the output is the same as the input as I had mentioned earlier. – Sreenath Feb 5 '18 at 12:40
• You try to solve a nonlinear pde,which doesn't contain derivatives D[Ze,t]. The output (same as input) -message shows that MMA doesn't find a solution... – Ulrich Neumann Feb 5 '18 at 14:42

If you solve your pde for D[Ze[X,t],X] Mathematica is able to solve the explicit pde:

tmp = Solve[pde == 0, D[Ze[X, t], X]][];
pdenew = tmp /. Rule -> Equal;
Simplify[ DSolve[pdenew, Ze[X, t], {X, t}][] ]
(*{Ze[X, t] ->E^((I w X Sec[theta])/ua) (-((2 I c E^((
I w (-2 c t ua Cos[theta] +
X (-2 c + ua + ua Cos[2 theta])) Sec[theta])/(2 c ua))
uat (1 + (E^((I w (-c t + X Cos[theta]))/c) uat)/ua)^(-((
c Sec[theta]^2)/ua))
Hypergeometric2F1[((-2 c + ua + ua Cos[2 theta]) Sec[
theta]^2)/(2 ua),
1 - (c Sec[theta]^2)/
ua, ((-c + ua + ua Cos[2 theta]) Sec[theta]^2)/
ua, -((E^((I w (-c t + X Cos[theta]))/c) uat)/ua)] Sin[
theta])/(w (-2 c + ua + ua Cos[2 theta]))) + (E^(I t w)
ua + E^((I w X Cos[theta])/c) uat)^(-((c Sec[theta]^2)/ua))
C[t])}*)

• @ Ulrich Could you please tell me which version of mathematica you use? – Sreenath Feb 5 '18 at 15:43
• @Sreenath: v 11.0.1 Windows 7 – Ulrich Neumann Feb 5 '18 at 21:23