Answer in Chemical Engineering for pavani #231347

7)Find the integral surface of x(y²+z)p-y(x²+z)q=(x²-y²)z through the curve x+y=0,z=1.

Auxiliary equations are

“{dxover x(y^2+z)}={dy over -y(x^2+z)}={dz over z(x^2-y^2)}”

By Choosing multipliers “x, y, -1,” we get

“{xdx+ydy-dzover x^2y^2+x^2z-x^2y^2-y^2z-x^2z+y^2z}={xdx+ydy-dzover 0}”

Then

“x^2+y^2-2z=C_1”

By Choosing multipliers “1/x, 1/y, 1/z,” we get

“{{dx over x}+{dx over x}+{dz over z}over y^2+z-x^2-z+x^2-y^2}={{dx over x}+{dx over x}+{dz over z}over 0}”

Then

“ln(xyz)=ln (C_2)”

Or

“xyz=C_2”

Parametric equation of straight line is

“x=t, y=-t, z=1”

Substitute

“t^2+(-t)^2-2(1)=C_1”“t(-t)(1)=C_2”

Eliminate “t”

“2t^2-2=C_1”“t^2=-C_2”

Then

“-2C_2-2=C_1”

Or

“C_1+2C_2+2=0”

Hence, the integral surface, which contains the straight line

“x^2+y^2-2z+2xyz+2=0”