(SINY-YSINX) dx+(COSX+XCOSY-Y) dy=0

(SINY-YSINX) dx+(COSX+XCOSY-Y) dy=0
in Calculus Answers by Level 12 User (101k points)

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15 Answers

Let P_=SINY-Y•SINX
by Level 12 User (101k points)
And Q_= COSX+XCOSY-Y
by Level 12 User (101k points)
Then æp/æy= COSY-SINX,
by Level 12 User (101k points)
æQ/æx=-SINX+COSY,
by Level 12 User (101k points)
P,y-Qx=0
by Level 12 User (101k points)
Therefore the given equation is exact :
by Level 12 User (101k points)
du_=Pdx+Qdy=0
by Level 12 User (101k points)
→ u(x,y)=c1
by Level 12 User (101k points)
æu/æy=Q
by Level 12 User (101k points)
U=y•COSX+X•SINY-y^2/2+w(x)
by Level 12 User (101k points)
æu/æx=-ySINX+SINY+w'(x)_=P
by Level 12 User (101k points)
→ w'(x)=0
by Level 12 User (101k points)
→ w(x)=c2
by Level 12 User (101k points)
That is, y•COSX+X•SINY-Y^2/2+c2=c1
by Level 12 User (101k points)
And finally: 2yCOSX+2x SINY=y^2+c
by Level 12 User (101k points)

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