Cfrgtkky

Find complete integral for partial differential equation $(y-x)(qy-px) = (p-q)^{2}$ where $p={ partial z over partial x},q={ partial z over partial y}$.

My attempt:

The given equation is f(x,y,z,p,q) = $(y-x)(qy-px) - (p-q)^{2}$
The Charpit's auxilary equation will be given by

$${dp over 2px-(p+q)y}={dq over 2qy-(p+q)x}={dx over {2(p-q)-x2+xy}}={dy over {-2(p-q)+xy-y2}}$$
From here I tried to proceed but no solvable fraction turned out.

Hint:

Let $begin{cases}u=x+y\v=x-yend{cases}$ ,

Then $dfrac{partial z}{partial x}=dfrac{partial z}{partial u}dfrac{partial u}{partial x}+dfrac{partial z}{partial v}dfrac{partial v}{partial x}=dfrac{partial z}{partial u}+dfrac{partial z}{partial v}$

$dfrac{partial z}{partial y}=dfrac{partial z}{partial u}dfrac{partial u}{partial y}+dfrac{partial z}{partial v}dfrac{partial v}{partial y}=dfrac{partial z}{partial u}-dfrac{partial z}{partial v}$

$therefore vleft(vdfrac{partial z}{partial u}+udfrac{partial z}{partial v}right)=4left(dfrac{partial z}{partial v}right)^2$

Take $X$ and $Y$ to be new variables such that

$$X=x+ytag1$$$$Y=xytag2$$

Given that $(y-x)(qy-px)=(p-q)^2$

Now, $$p=dfrac{partial z}{partial x}=dfrac{partial z}{partial X}dfrac{partial X}{partial x}+dfrac{partial z}{partial Y}dfrac{partial Y}{partial x}=dfrac{partial z}{partial X}+ydfrac{partial z}{partial Y}tag3$$and
$$q=dfrac{partial z}{partial y}=dfrac{partial z}{partial X}dfrac{partial X}{partial y}+dfrac{partial z}{partial Y}dfrac{partial Y}{partial y}=dfrac{partial z}{partial X}+xdfrac{partial z}{partial Y}tag4$$

Now plug in $p$ and $q$ in $(2)$ and we get
$$(y-x)left[yleft(dfrac{partial z}{partial X}+xdfrac{partial z}{partial Y}right)-xleft(dfrac{partial z}{partial X}+ydfrac{partial z}{partial Y}right)right]=left[left(dfrac{partial z}{partial X}+ydfrac{partial z}{partial Y}right)-left(dfrac{partial z}{partial X}+xdfrac{partial z}{partial Y}right)right]^2$$or
$$implies(y-x)^2dfrac{partial z}{partial X}=(y-x)^2left(dfrac{partial z}{partial Y}right)$$
$$implies dfrac{partial z}{partial X}=left(dfrac{partial z}{partial Y}right)^2mbox{ or } p=q^2tag5$$
where $p=dfrac{partial z}{partial X}$ and $q=dfrac{partial z}{partial Y}$

$(4)$ is of the form $f(p,q)=0$

So, $$z=aX+bY+ctag6$$

where $a=b^2$, on putting $a$ for $p$ and $b$ for $q$ in $(5)$

Therefore, from $(6)$, the required complete integral is $$z=b^2X+bY+c$$or$$z=b^2(x+y)+bxy+c mbox{ from }(6)$$