Play with Curves!

$\begin{aligned} \frac d{dx} \left( \frac{x^a}{b} + \frac{x^c}{d} - exy - f \cos (gy) + hx \ln (iy) \right) & = 0 \\ \frac{ax^{a-1}}{b} + \frac{cx^{c-1}}{d} - ey - ex \frac {dy}{dx} + gf \sin (gy) \frac {dy}{dx} + h\ln (iy) + \frac {ihx}{iy} \frac {dy}{dx} & = 0 \end{aligned}$

$\begin{aligned} \implies \frac {dy}{dx} & = \frac {ey - \frac{ax^{a-1}}{b} - \frac{cx^{c-1}}{d} - h\ln (iy)}{gf \sin (gy) - ex - \frac {hx}{y}} = \frac{ y - x^3 - x^2 - \ln y }{\sin y - x + \frac{x}{y}} \end{aligned}$

Comparing the coefficients, we have:

$\implies a=b=4$ , $c=d=3$ , and $e=f=g=h=i=1$ , therefore, $a+b+c+d+e+f+g+h+i =$ $4+4+3+3+1+1+1+1+1=\boxed{19}$ .

Its a simple variable separation method.

Cross mutliply

to get

d(x) denotes derivative

d(xy)-(x^3)dx-(x^2)dx = sinydy +d(xlny)

Integrate directly to get required equation

xy - x^4 / 4 - x^3 / 3 = -cosy +xlny