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How can I input this differential equation in Mathematica and see the solving steps?

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If we assume that the solution is equally valid using Mathematica and Wolfram Alpha, I see no reason to close this question. There must be some sensible threshold before the knee-jerk close question reaction just because a webapp was mentioned. – fideli Nov 14 '10 at 20:41
Oh, my bad, I figured Mathematica was part of Wolfram Alpha. As apparently it's not, is just a bad question for any Q&A site, it being too broad? Still then, migrating is not closing, but a way to help folks getting better answers. – Arjan Nov 14 '10 at 20:51
I don't see any reason why it shouldn't be here, but might be better. Wolfram Alpha has notoriously poor input functionality. – Dennis Williamson Nov 14 '10 at 21:32
If you want to ask about web apps, try otherwise, leave just the non-webapps portion – random Nov 15 '10 at 4:24
y= -x is a trivial solution – Dr. belisarius Feb 14 '11 at 15:55

In Mathematica, you need to put it in the following form:

eqn = x*y'[x] - y[x] == Sqrt[1 + x^2]*y'[x] + Sqrt[1 + y[x]^2]
DSolve[eqn, y[x], x]

With that I get:

{{y[x] -> InverseFunction[1/2 (ArcSinh[#1] + #1 (-#1 + Sqrt[1 + #1^2])) &][ 1/2 (-x (x + Sqrt[1 + x^2]) - ArcSinh[x]) + C[1]]}}

with the following warning:

Solve::tdep: The equations appear to involve the variables to be solved for in an essentially non-algebraic way.

From the help it appears that the solver encountered a transcendental function. That's about as far as I can take this. Hope this helps get you going.

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