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

Posted 8 years ago
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 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.
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Posted 8 years ago
 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:= eqn = y'''[x]*(1 + y'[x]^2) == 3*y'[x]*y''[x]^2; In:= (sol = DSolve[eqn, y, x]) // InputForm Out//InputForm= {{y -> Function[{x}, ((-I)*Sqrt[-1 + x^2*C^2 + 2*x*C^2*C + C^2*C^2])/C + C]}, {y -> Function[{x}, (I*Sqrt[-1 + x^2*C^2 + 2*x*C^2*C + C^2*C^2])/C + C]}} In:= eqn /. sol // Simplify Out= {True, True} In:= implicitsol = (x^2 + y[x]^2 + c*x + d*y[x] + e == 0); In:= Simplify[implicitsol /. sol /. {c -> 2*C, d -> -2*C, e -> (-1 + C^2*C^2 + C^2*C^2)/C^2}] Out= {True, True} 
Posted 8 years ago
 In:= Timing @ DSolve[y'''[x]*(1 + y'[x]^2)^(3/2) == 3*y'[x]*y''[x]^2, y[x], x] returns unevaluated after 240 seconds
Posted 8 years ago
 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:http://reference.wolfram.com/language/ref/DSolve.htmlIf 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/)?