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

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.

Mathematicaand 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 20:41notclosing, but a way to help folks getting better answers. – Arjan Nov 14 '10 at 20:51mightbe better. Wolfram Alpha hasnotoriouslypoor input functionality. – Paused until further notice. Nov 14 '10 at 21:32