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Calculus Level 5

The general solution of the differential equation \(({ y }^{ 2 }+2{ t }^{ 2 }y)dt+(2{ t }^{ 3 }-ty)dy=0\) is of the form: \(a(ty)^{ { c }/{ d } }-\dfrac { e }{ f } \left(\dfrac { y }{ t } \right)^{ { g }/{ h } }=A\), where \(A\) is an arbitrary constant and \(a,c,d,e,f,g,h\) are all natural numbers, \(a\) is not divisible by 3, and \(a + e < 14.\)

Find \(a+d+e+f+g+h\).

Note: c,d are coprime,
e, f are prime,
g, h are coprime.

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