Advertisements
Advertisements
Question
Let u = x2y3 cos`(x/y)`. By using Euler’s theorem show that `x*(del"u")/(delx) + y * (del"u")/(dely)`
Advertisements
Solution
Given, u = x2y3 cos`(x/y)`
i.e., u(tx, ty) = (tx)2 (ty)3 cos`("tx"/"ty")`
= t2x2t3y3 cos`(x/y)`
= t5x2y3 cos`(x/y)`
= t5 u
∴ u is a homogeneous function in x and y of degree 5.
∴ By Euler’s theorem, `x * (del"u")/(delx) + y * (del"u")/(dely)` = 5u
Hence Proved.
APPEARS IN
RELATED QUESTIONS
Let u = x cos y + y cos x. Verify `(del^2"u")/(delxdely) = (del^"u")/(del"y"del"x")`
If q = 1000 + 8p1 – p2 then, `(del"q")/(del "p"_1)`is:
Find the partial dervatives of the following functions at indicated points.
f(x, y) = 3x2 – 2xy + y2 + 5x + 2, (2, – 5)
Find the partial derivatives of the following functions at indicated points.
h(x, y, z) = x sin (xy) + z2x, `(2, pi/4, 1)`
For the following functions find the fx, and fy and show that fxy = fyx
f(x, y) = `cos(x^2 - 3xy)`
For the following functions find the gxy, gxx, gyy and gyx
g(x, y) = log(5x + 3y)
If V(x, y) = ex (x cosy – y siny), then Prove that `(del^2"V")/(delx^2) + (del^2"V")/(dely^2)` = 0
If w(x, y) = xy + sin(xy), then Prove that `(del^2w)/(delydelx) = (del^2w)/(delxdely)`
If v(x, y, z) = x3 + y3 + z3 + 3xyz, Show that `(del^2"v")/(delydelz) = (del^2"v")/(delzdely)`
A from produces two types of calculates each week, x number of type A and y number of type B. The weekly revenue and cost functions = (in rupees) are R(x, y) = 80x + 90y + 0.04xy – 0.05x2 – 0.05y2 and C(x, y) = 8x + 6y + 2000 respectively. Find `(del"P")/(delx)` (1200, 1800) and `(del"P")/(dely)` (1200, 1800) and interpret these results
