Play with Curves!

Calculus Level 4

\[ \frac{dy}{dx} = \frac{ y - x^3 - x^2 - \ln y }{\sin y - x + \frac{x}{y}} \]

Given that the slope of a curve passing through \( ( 0 , \frac{\pi}{2} ) \) is as shown above.

The equation of curve can be represented as

\[\frac{x^a}{b} + \frac{x^c}{d} - exy - f \cos (gy) + hx \ln (iy) = 0, \]

where \(a\), \(b\), \(c\), \(d\), \(e\), \(f\), \(g\), \(h\), and \(i\) are positive integers.

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


Original.

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