How to solve a 3th order non-linear differential equation?

Questions
,
1

I want to find, with basic method, the general solutions of the following equation$$
y’‘’(1+y’^2)^\frac{3}{2}=3y’(y’')^2.

The answer of this one is short and without any suggestion: $$ x^2+y^2+cx+dy+e=0$$ I tried to use Mathematica to solve this, but I failed. I don't meet anywhere, in examples or something else, a such of equations. Thank you and I apologize if I missed something.
2

Are you using Mathematica or Wolfram|Alpha?

The first step is write out the equation in Mathematica. Here are some examples of how to solve differential equations in Mathematica:

If you have trouble with the syntax, can you enter the equation into Wolfram|Alpha? Have you tried entering the equation into Mathematica with Free Form Input (https://www.wolfram.com/broadcast/screencasts/free-form-input/)?

3

In[1]:= Timing @
DSolve[y’‘’(1 + y’[x]^2)^(3/2) == 3y’*y’'^2, y, x]

returns unevaluated after 240 seconds

4

DSolve attempts to solve this nonlinear ODE by reducing the problem to a second-order equation,
but is unable to find a closed-form solution.

It may be noted that the following simpler problem can be solved quickly and leads to a solution similar
to that suggested by you:

In[1]:= eqn = y'''[x]*(1 + y'[x]^2) == 3*y'[x]*y''[x]^2;

In[2]:= (sol = DSolve[eqn, y, x]) // InputForm

Out[2]//InputForm=
{{y -> Function[{x}, ((-I)*Sqrt[-1 + x^2*C[1]^2 + 2*x*C[1]^2*C[2] + C[1]^2*C[2]^2])/C[1] + C[3]]}, 
 {y -> Function[{x}, (I*Sqrt[-1 + x^2*C[1]^2 + 2*x*C[1]^2*C[2] + C[1]^2*C[2]^2])/C[1] + C[3]]}}

In[3]:= eqn /. sol // Simplify

Out[3]= {True, True}

In[4]:= implicitsol = (x^2 + y[x]^2 + c*x + d*y[x] + e == 0);

In[5]:= Simplify[implicitsol /. sol /. {c -> 2*C[2], d -> -2*C[3], 
   e -> (-1 + C[1]^2*C[2]^2 + C[1]^2*C[3]^2)/C[1]^2}]

Out[5]= {True, True}