Let u = x2 y3 cos(\(\frac{x}{y}\)). By using Euler’s theorem show th

1.	Let u = x2 y3 cos(\(\frac{x}{y}\)). By using Euler’s theorem show that \(x.\frac{∂u}{∂x}+y.\frac{∂u}{∂y}\)
Answer» Given, u = x² y³ cos(\(\frac{x}{y}\)) i.e., u(tx, ty) = (tx)² (ty)³ cos\((\frac{tx}{ty})\) = t² x² t³ y³ cos(\(\frac{x}{y}\)) = t⁵x² y³ cos(\(\frac{x}{y}\)) = t⁵ u ∴ u is a homogeneous function in x and y of degree 5. ∴ By Euler’s theorem, \(x.\frac{∂u}{∂x}+y.\frac{∂u}{∂y}\) = 5u Hence Proved.

Let u = x2 y3 cos(\(\frac{x}{y}\)). By using Euler’s theorem show that \(x.\frac{∂u}{∂x}+y.\frac{∂u}{∂y}\)

Answer»

Given, u = x² y³ cos(\(\frac{x}{y}\))

i.e., u(tx, ty) = (tx)² (ty)³ cos\((\frac{tx}{ty})\)

= t² x² t³ y³ cos(\(\frac{x}{y}\))

= t⁵x² y³ cos(\(\frac{x}{y}\))

= t⁵ u

∴ u is a homogeneous function in x and y of degree 5.

∴ By Euler’s theorem, \(x.\frac{∂u}{∂x}+y.\frac{∂u}{∂y}\) = 5u

Hence Proved.