For an equation of the form $\displaystyle f\left(\frac{\partial z}{\partial x}, \frac{\partial z}{\partial y}\right) = 0$ the complete solution is $ z = ax + \phi (a) y + \psi(a)$--(1) and general solution is given by eliminating a between (1) and $\displaystyle \frac{\partial z}{\partial a} = 0$. I don't get it how. Can anyone help me by giving me an example with particular problem below?

Let, $3p^2 - 2 q^2 = 4pq$ where $p$ and $q$ are $x$ and $y$ partial derivatives of $z$ respectively. The complete solution is given by $\displaystyle z = a \left( x + \frac{3}{ 2 \pm \sqrt{10}} y\right) + k$. How to find general solution for this particular problem?

  • $\begingroup$ The first sentence doesn't make sense: are there additional conditions on the function $f$ that you didn't tell us? $\endgroup$ – Willie Wong Sep 3 '12 at 13:23
  • $\begingroup$ Also, if this is coming from a textbook, can you state which textbook you are using, and if possible include how the textbook/lecture notes/paper defines "complete solution" and "general solution"? $\endgroup$ – Willie Wong Sep 3 '12 at 13:25

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