Solving Implicit O.D.E: sin(y) + xy - x^3 = 2 | 2nd Order ODE Help

[AST3019]

Is, sin(y) + xy - x^3=2, an implicit soln to the 2nd order ODE y''= {6xy' + (y')^3 * sin(y) - 2(y')^2}/ (3x^2 - y)?

 

[HallsofIvy]

Have you tried it? If $sin(y) + xy - x^3=2$ then $cos(y)y'+ y+ xy'- 3x^2= 0$. Differentiating again, $-sin(y)(y')^2+ cos(y)y''+ 2y'+ xy''- 6x= 0$.

You can you those to see if the differential equation is satisfied or not.

 

[AST3019]

thanks.