Question

1. Form the partial differential equation by eliminating
the arbitrary constants a and b from z=ax+by.
2. Form the partial differential equation by eliminating
the arbitrary constants a and b from z=ax

Answer 1

$\text{1. Given \( z = ax + by\) then we will have the following: } \\ \displaystyle \frac{\partial z}{\partial x} = a \text{ and } \displaystyle \frac{\partial z}{\partial y} = b\\ \text{Inserting these into \(z = ax + by\) we have}\\ \displaystyle z = x\frac{\partial z}{\partial x} +y\frac{\partial z}{\partial y} \\ \text{which is equivalent to: } \\ x\frac{\partial z}{\partial x} +y\frac{\partial z}{\partial y} - z = 0 \\ \text{2. Given \( z = ax \) then we will have the following: } \\ \displaystyle \frac{\partial z}{\partial x} = a \\ \text{Inserting these into \(z = ax\) we have}\\ \displaystyle z = x\frac{\partial z}{\partial x}$