Consider the equation $x^2p^2 + y^2q^2=u$. Do the transformation $(x,y,u) → (\bar x,\bar y,\bar u)$ defined by $$\bar x = \ln x, \bar y = \ln y, \bar u = u$$ and obtain an equation involving only $\bar p$ and $\bar q$. Find a complete integral of this equation and express the final result in terms of the original variables $x,y,u.$
First I did the transformation.
$$F=x^2p^2 + y^2q^2-u=0$$
$$p=u_x= u_\bar x \bar x_x + u_\bar y \bar y_x= u_\bar x /x$$
$$q=u_y= u_\bar x \bar x_y + u_\bar y \bar y_y= u_\bar y /y$$
$$F=0=x^2p^2+y^2q^2-u= x^2(u_\bar x /x)^2+y^2(u_\bar y /y)^2-u$$
$$0=u_\bar x^2+u_\bar y^2 -u$$
$$0=\bar p^2 + \bar q^2-u$$
But I am confused because question is like that "obtain an equation involving only $\bar p$ and $\bar q$." But I also have $u$ in my equation. What is my mistake? If someone solves my question, I would appreciate. Thanks!