Question

Find the integral surface of the equation 4yzp+q+2y=0 passing through

y²+z²=2,x+z=1

Answer 1

$\\\text{Given:}\ 4 y z p+q+2y=0 \ldots (1) \\\text{Given curve is}\ y^{2}+z^{2}=2, x+z=1 \ldots (2) \\\text{The Lagrange’s auxiliary equations of (1) are} \\\frac{d x}{4 y z}=\frac{d y}{1}=\frac{d z}{-2 y} \quad \ldots (3) \\\text{Taking the first and third fraction of (3), we have} \\d x+2 z d z=0, \text{so that}\ x+z^2=c_{1} \cdots (4) \\\text{Taking the last two fractions of (3), we have} \\d z+2 y d y=0 \text { so that }\ z+y^2=c_{2} \ldots (5) \\\text{Adding (4) and (5),} \\ x+z^2+ z+y^2= c_{1}+ c_{2} \\\Rightarrow y^{2}+z^{2}+x+z=c_{1}+c_{2} \\\Rightarrow 2+1=c_{1}+c_{2} \\\Rightarrow c_{1}+c_{2} =3 \\\text{The equation of required surface is} \\x+z^2+ z+y^2 =3 \\\Rightarrow y^{2}+z^{2}+x+z-3=0$