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Équation différentielle de Madame Ratte

BigRat
BigRat used Ask the Experts™
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On cherche la solution de l'équation :-

(xy''' - y'')² = (y''')² + 1

C'est clair, non?


[Les fromages vont aller à la solution la plus élégante]

[y''' = d³y/dx³, y'' = d²y/dx² ]
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Top Expert 2006
Commented:
Let z=y'', then it simplifies to only a first-order nonlinear ODE:

   (x.z' - z)^2 = (z' )^2 + 1

Trial a linear solution for z = Ax+K, and you find it is indeed a solution which results in the condition

   K^2 - A^2 = 1.

Then integrate twice to get the y solution:

   y(x) = 1/6*A*x^3 + 1/2*sqrt(A^2+1)*x^2 + Bx + C.

Fromage pour moi?

Author

Commented:
>>Fromage pour moi?

Mais certainement!

Jus one small question: Why was the question in French? And what is the clue to solve the equation?

Top Expert 2006

Commented:
The question was in French because you are French? And...I didn't pick up on any clues per se (are there any?); the steps I took seemed fairly obvious in and of themselves (although having said that, I did while away nearly an hour playing with Taylor expansions).

Author

Commented:
If you do the substitution for y'' as z for example and rearange the equation you'll get it in the form :-

     z = xz' + f(z')

where the f(z') is a function of z' only; in this case sqrt(z'²+1).

This is Clairaut's equation whose solution is to replace z' by a constant.
http://mathworld.wolfram.com/ClairautsDifferentialEquation.html

Thereafter integrating twice gives the result.

The clue was the French phrase "C'est clair, non" meaning that it is in fact a Clairaut equation.

>>The question was in French because you are French?

Everything south of Flanders is not French. I'm originally from Luxembourg.

Top Expert 2006

Commented:
Interesting, I was not aware of Clairaut's DE. Thank you!