Question : Équation différentielle de Madame Ratte

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 ]

Answer : Équation différentielle de Madame Ratte

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?