The auxiliary equation is x2dx=y2dy=−z2dz.
Taking the first two ratios,
x2dxIntegrating both sides, we getx−1y1−x1=y2dy=y−1+c1=c1(1)
Taking last two ratios,
y2dyIntegrating both sides, we gety−1−y1−z1=−z2dz=z1+c2=c2(2)
The integral curve is ϕ(c1,c2)=0, i.e., ϕ(y1−x1,−y1−z1)=0.
Given that the integral surface passes through xy=x+y,z=1.
Letting x(s)=s as the parametric representation, we get
sysy−yy(s−1)∴y(s)=s+y=s=s=s−1s
Using x(s)=s,y(s)=s−1s,z=1 in (1) and (2) we get
ss−1−s1−(ss−1)−1=c1⇒c1=ss−2⇒c1=1−s2⇒s1=21−c1(3)=c2⇒c2=s1−2s⇒c2=s1−2(4)
Using (3) in (4),
c22c2−1+c1+4c1+2c2+3(y1−x1)−2(y1+z1)+3xyx−y−2(yzz+y)+3xz−yz−2xz−2xy+3xyz3xyz−2xy−xz−yz=21−c1−2=0=0=0 (Using (1) & (2))=0=0=0
which is the required integral surface.
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