solving differential equations of the following form

Question

I understand that the questions are from the 'Differential Equations' part of the syllabus. It also seems that p means $dy/dx$ in these types. But I am not sure about the rest of the notations and also what exactly in 'differential equations' topic do these relate to. Most of the questions I have encountered do not have other letters apart from p,x and y. Any simple reference would be much appreciated.

• $px+qy=pq$
• $pt-qs=q^3$
• $pxy+pq+qy=yz$
• $pq=x(ps-qr)$

Answer 1

HINT to begin:

When $z=f(x,y)$, partial derivatives of z are
$$ p=\dfrac{\partial z}{\partial x} ,\,q=\dfrac{\partial z}{\partial y} ,\, r=\dfrac{\partial^2 z}{\partial x^2} ,\,t=\dfrac{\partial^2 z}{\partial y^2} ,\,s=\dfrac{\partial^2 z}{\partial x\, \partial y} . $$

Answer 2

These are all rather badly [b]non[/]-linear equations so do not have any simple solutions. And the third is not even an equation!