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$p=\dfrac{dz}{dx}$ and $q=\dfrac{dz}{dy}$

I have to find the complete integral solution of the Partial Differential Equation

$$xp(1+q)=(y+z)q$$

$$ \Rightarrow f(x,y,z,p,q) = xp + xpq – yq – zq = 0$$

I am trying to solve it using Charpit equations :

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

For the first and the last fraction: $f_p=x(1+q)$ and $f_y+pf_z=-q(1+q)$; which gives- $$\dfrac{dx}{-x(1+q)}=\dfrac{dq}{-q(1+q)}\Rightarrow \log(x) + \log(a) = \log(q)\Rightarrow q=ax.$$

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From the given PDE and $q=ax$, we have $p=\dfrac{a(y+z)}{1+ax}.$

Now for the solution, we have to solve $dz=pdx+qdy \Rightarrow dz=\dfrac{a(y+z)}{1+ax}dx + axdy.$

I am not able to integrate $dz=\dfrac{a(y+z)}{1+ax}dx + axdy$. Any ideas on how to integrate this to get expression for $z$ in terms of $x$ and $y$?

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Supposing that $q=ax$ and $p=\frac{a(y+z)}{1+ax}$ (I didn’t check the simajinid’calculus).

$$q=\frac{\partial z}{\partial y}=ax\quad\to\quad z=axy+\phi(x)$$

$$p=\frac{\partial z}{\partial x}=ay+\phi'(x)=\frac{a(y+axy+\phi(x))}{1+ax}$$

$$\phi'(x)=\frac{a\:\phi(x)}{1+ax}$$

$$\phi(x)=b(1+ax)$$

$$z(x,y)=axy+b(1+ax)$$

Of couse, this is only one particular solution, not the general solution of the PDE.

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