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

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SUMMARY

The discussion focuses on determining whether the implicit solution sin(y) + xy - x^3 = 2 satisfies the second-order ordinary differential equation (ODE) y'' = {6xy' + (y')^3 * sin(y) - 2(y')^2} / (3x^2 - y). Participants confirm that differentiating the implicit solution leads to the equation cos(y)y' + y + xy' - 3x^2 = 0, which can be further differentiated to yield -sin(y)(y')^2 + cos(y)y'' + 2y' + xy'' - 6x = 0. This confirms the relationship between the implicit solution and the ODE.

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

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

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