how do you solve (siny-ysinx)dx+(cosx+xcosy-y)dy=0

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16 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-Q'x=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 :
by Level 12 User (101k points)
2yCOSX + 2xSINY= y^2+ c
by Level 12 User (101k points)

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