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

Posted 8 years ago
3 Replies
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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.

POSTED BY: teo ytriu
3 Replies

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

{{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}
POSTED BY: Devendra Kapadia
In[1]:= 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 BY: Frank Kampas

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 (

POSTED BY: Sean Clarke
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