Q.1 (c) If u=(1-2xy+y²)^(-1/2) , prove that x∂ u/∂ x-y∂ u/∂ y=y²u³ .

∂ u/∂ x=-1/2(1-2xy+y²)^(-3/2)(-2y)=y u³ ∂ u/∂ y=-1/2(1-2xy+y²)^(-3/2)(-2x+2y)=(x-y) u³ x∂ u/∂ x-y∂ u/∂ y=xy u³-(xy-y²)u³ =y²u³ (proved).

If \(u=(1-2xy+y^2)^{-1/2}\), prove that \(x\frac{\partial u}{\partial x}-y\frac{\partial u}{\partial y}=y^2u^3\).

Q.1 (c) If \(u=(1-2xy+y^2)^{-1/2}\), prove that \(x\frac{\partial u}{\partial x}-y\frac{\partial u}{\partial y}=y^2u^3\).

Doubtly Asked 25 Oct 2026 mu engineering Save WhatsApp Report

Best answer

\(\frac{\partial u}{\partial x}=-\frac12(1-2xy+y^2)^{-3/2}(-2y)=y\,u^3\)

\(\frac{\partial u}{\partial y}=-\frac12(1-2xy+y^2)^{-3/2}(-2x+2y)=(x-y)\,u^3\)

\(x\frac{\partial u}{\partial x}-y\frac{\partial u}{\partial y}=xy\,u^3-(xy-y^2)u^3\)

\(=y^2u^3\) (proved).

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

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