Question

Z=a(x+log y)-x^2/2-bx

Answer 1

$$z(x,y)=a(x+\log y)-\frac{x^2}{2}-bx\\ z= ax +a\log y-\frac{x^2}{2}-bx$$

for z(x,y):

$$dz = \dfrac{\partial z}{\partial x}dx+\dfrac{\partial z}{\partial y}dy\\$$

then;

$$\dfrac{\partial z}{\partial x} = a - \frac{x^3}{6}-b\; \text{;} \qquad \dfrac{\partial z}{\partial y} = \frac{a}{y}$$

Let:

$$p = \dfrac{\partial z}{\partial x} \qquad \text{and } \qquad q=\dfrac{\partial z}{\partial y}$$

then

$$q = \frac{a}{y} \implies a =qy \quad \cdots (i)\\ p = a -\frac{x^3}{6}-b \qquad \cdots(ii)$$

put (i) in (ii)

$$p = qy -\frac{x^3}{6}-b\\ 6p = 6qy - x^3 - 6b\\ 6p-6qy+x^3+6b=0\\ 6(p-qy)+x^3+6b=0\\ 6(p-qy+b)+x^3=0\\$$

The required PDE is therefore:

$$6(p-qy+b)+x^3=0\\$$