Question

Form a partial differential equation by eliminating the function f from z = f(y/x)

Answer 1

Given, $z = f\left(\dfrac{y}{x}\right)$.

Differentiating partially with respect to $x$, we get

$p=\dfrac{\partial z}{ \partial x}=f'\left(\dfrac{y}{x}\right)\cdot \dfrac{-y}{x^2}\\ \\ \therefore~f'\left(\dfrac{y}{x}\right)= -\dfrac{px^2}{y} \qquad\qquad~~~~~(1)$

Differentiating partially with respect to $y$, we get

$q=\dfrac{\partial z}{ \partial y}=f'\left(\dfrac{y}{x}\right)\cdot \dfrac{1}{x}\\ f'\left(\dfrac{y}{x}\right) = qx \qquad\qquad\qquad~~~~~~~~(2)$

From (1) and (2), we get

$\begin{aligned} -\dfrac{px^2}{y} &= qx\\ -px &= qy\\ xp + yq &= 0 \end{aligned}$

which is the required partial differential equation.