Q.1 (a) If u=sqrt x+sqrt y/x+y find x∂ u/∂ x+y∂ u/∂ y .

u(tx,ty)=sqrt t(sqrt x+sqrt y)/t(x+y)=t^(-1/2)u , so u is homogeneous of degree n=-1/2 . By Euler's theorem xu_x+yu_y=nu : x∂ u/∂ x+y∂ u/∂ y=-1/2u=-sqrt x+sqrt y/2(x+y)

If \(u=\frac{\sqrt x+\sqrt y}{x+y}\) find \(x\frac{\partial u}{\partial x}+y\frac{\partial u}{\partial y}\).

Q.1 (a) If \(u=\frac{\sqrt x+\sqrt y}{x+y}\) find \(x\frac{\partial u}{\partial x}+y\frac{\partial u}{\partial y}\).

Doubtly Asked 25 Oct 2026 mu engineering Save WhatsApp Report

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\(u(tx,ty)=\frac{\sqrt t(\sqrt x+\sqrt y)}{t(x+y)}=t^{-1/2}u\), so u is homogeneous of degree \(n=-\frac12\).

By Euler's theorem \(xu_x+yu_y=nu\):

\(x\frac{\partial u}{\partial x}+y\frac{\partial u}{\partial y}=-\frac12u=-\frac{\sqrt x+\sqrt y}{2(x+y)}\)

Doubtly AI · 25 Oct 2026 AI-verified · Oct 2026 Report

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If \(u=\frac{\sqrt x+\sqrt y}{x+y}\) find \(x\frac{\partial u}{\partial x}+y\frac{\partial u}{\partial y}\).

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