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Équation différentielle de Madame Ratte

BigRat used Ask the Experts™
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
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?


>>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

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).


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.

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

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