Determine the equation of the curve passing through the origin, in the form of \( y=f(x) \) , which satisfies the differential equation \( \frac{\mathrm{dy}}{\mathrm{dx}}=\sin (10 \mathrm{x}+6 \mathrm{y}) \) .
(A) \( 5 x+3 y=\left[\tan ^{-1}\left(\frac{5 \tan 4 x}{4-3 \tan 4 x}\right)-5 x\right] \)
(B) \( 5 x+3 y=\left[\tan ^{-1}\left(\frac{5 \tan 4 x}{4-3 \tan 4 x}\right)+7 x\right] \)
(C) \( 5 x-3 y=\left[\tan ^{-1}\left(\frac{5 \tan 4 x}{4-3 \tan 4 x}\right)-8 x\right] \)
(D) \( 9 x+7 y=\left[\tan ^{-1}\left(\frac{5 \tan 4 x}{4-3 \tan 4 x}\right)-5 x\right] \)