A not so easy differential equation

Calculus Level 5

dydxd3ydx3=3(d2ydx2)2\dfrac { dy }{ dx } \cdot \dfrac { { d }^{ 3 }y }{ d{ x }^{ 3 } } =3{ \left( \dfrac { { d }^{ 2 }y }{ d{ x }^{ 2 } } \right) }^{ 2 } The general solution of the differential equation above is of the form yα=kx+fβg+c{ y }^{ \alpha }=\dfrac { \sqrt [ \beta ]{ kx+f } }{ g } +c, where k,k, f,f, g,g, and cc are arbitrary constants and α\alpha and β\beta are coprime positive integers.

Find the value of  α+β.\ \alpha +\beta.

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