The given equation is-
x(y2−a2)dx+y(x2−t2)dy−z(y2−α2)dz=0,
Given equation is of the form-
Pp+Qq+Rr=0
The Pdx=Qdy=Rdz
So,x(y2−a2)dx=y(x2−t2)dy=z(y2−α2)dz
Taking first two terms-
x(y2−a2)dx=y(x2−t2)dy
⇒x(x2−t2)dx=y(y2−a2)dy
Integrating both the sides-
2x2−t2lnx=2y2−a2lny+c1
⇒2x2−t2lnx−2y2+a2lny=c1 −(1)
Taing last two terms-
y(x2−t2)dy=z(y2−α2)dz
⇒y(y2−α2)dy=z(z2−t2)dz
Integrating Both the sides-
2y2−α2lny=2z2−t2lnz+c2
⇒2y2−α2lny−2z2+t2lnz=c2
Hence The solution is-
ϕ(c1,c2)=0
ϕ(2x2−t2lnx−2y2+a2lny,2y2−α2lny−2z2+t2lnz)=0
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