The general solution of the equation

\(x \frac{\partial z}{\partial x}+y \frac{\partial z}{\partial y}=0\) is

Explanation:

Given: \(x \frac{\partial z}{\partial x}+y \frac{\partial z}{\partial y}=0\) i.e. xp + yq = 0

on comping with Pp + Qq = R, we have-

P = x, Q = y & R = 0

so, By Lagrange auxillary equation

 \(\frac{d x}{p}=\frac{d y}{Q}=\frac{d z}{R}\)

\(\Rightarrow \frac{d x}{x}=\frac{d y}{y}=\frac{d z}{0}\)

Now dz = 0

⇒ z = c₁

using first and 2^nd term

 \(\frac{d x}{x}=\frac{d y}{y}\)

Integrating, 

log |z| = log |y| + log c₂

\(\Rightarrow \log \left(\frac{|x|}{|y|}\right)=\log c_2\)

\(\Rightarrow c_2=\frac{|x|}{|y|}\)

Hence, general sol is -

c₁ ϕ(c₂) or c₂ = ϕ(c₁) or ϕ(c₁ c₂) = o

⇒ z = ϕ \(\left(\frac{|x|}{|y|}\right)\) 

⇒ option (1) correct