Question

the characteristic of the equation PQ=x and the integral surface which passes through the curve y=1/2 z=x are =?

Answer 1

$$f(x, y, z, p, q)=pq-x=0$$ $$f_x=-1, f_y=0, f_z=0, f_p=q, f_q=p$$

$$\dfrac{dx}{f_p}=\dfrac{dy}{f_q}=\dfrac{dz}{pf_p+qf_q}=\dfrac{dp}{-(f_x+pf_z)}=\dfrac{dq}{-(f_y+qf_z)}$$

$\dfrac{dx}{q}=\dfrac{dy}{p}=\dfrac{dz}{2pq}=\dfrac{dp}{1}=\dfrac{dq}{0}$

$dx/q=dp$

$dq=0\implies q=c$

$p=x/c+c_1$

$z=x^2/(2c)+c_1x+c_3$

$dy=pdp$

$p^2=2y+c_2$

$(x/c+c_1)^2=2y+c_2$

$F(z-x^2/(2c)-c_1x,(x/c+c_1)^2-2y)=0$

we have y=1/2 and z=x, so

$F(x(1-c_1)-x^2/(2c),(x/c+c_1)^2-1)=0$