The given equation can be separated as, cosx dx=(y+2)(y+4)6y+16dy.
Now using partial fractions for the term on the right hand side,
(y+2)(y+4)6y+166y+16=y+2A+y+4B=A(y+4)+B(y+2)
Taking y=−4 gives, B=4 and taking y=−2 gives, A=2.
Therefore,
cosx dx={y+22+y+44}dy
Integrating both sides, we get
sinxsinxsinx+ce(sinx)+c(y+2)2(y+4)4−esinx=2log(y+2)+4log(y+4)=log(y+2)2+log(y+4)4=log{(y+2)2(y+4)4}=(y+2)2(y+4)4=c1 (where c1=ec)
Hence, F(x,y)=G(x)+H(y)=−esinx+(y+2)2(y+4)4 where
G(x)=−esinx;H(y)=(y+2)2(y+4)4.
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