Advertisements
Advertisements
प्रश्न
Prove that g(x, y) = `x log(y/x)` is homogeneous What is the degree? Verify Eulers Theorem for g
उत्तर
g(x, y) = `x log(y/x)`
g(tx, ty) = `"t"x log(("t"y)/("t"x))`
g is a homogeneous function of degree 1.
∴ By Euler’s Theorem,
`x (del"g")/(delx) + y (del"g")/(dely)` = g
Verification:
g(x, y) = `xlog(y/x)`
= `x (logy - log x)`
= `x log y - x log x`
`(del"g")/(delx) = logy - logx - x xx 1/x`
= `log y - log x - 1`
`x (del"g")/(delx) = x log y - x log x - x`
`(del"")/(dely) = x xx 1/y`
`y (delg")/(dely)` = x
`x (del"g")/(delx) + y (del"g")/(dely) = x log y - x log x - x + x`
= `x log (y/x)`
= g
`x (del"g")/(delx) + y (del"g")/(dely)` = g
Hence verified.
APPEARS IN
संबंधित प्रश्न
Let u(x, y, z) = xy2z3 x = sin t, y = cos t, z = 1 + e2t, Find `"du"/"dt"`
Let U(x, y, z) = xyz, x = e–t, y = e–t cos t, z – sin t, t ∈ R, find `"dU"/"dt"`
Let z(x, y) = x tan–1(xy), x = t², y = s et, s, t ∈ R. Find `(delz)/(del"s")` and `(delz)/(del"t")` at s = t = 1
Let U(x, y) = ex sin y where x = st2, y = s2t, s, t ∈ R. Find `(del"U")/(del"s"), (del"u")/(del"t")` and evaluate them at s = t = 1
Let z(x, y) = x3 – 3x2y3 where x = set, y = se–t, s, t ∈ R. Find `(delz)/(del"s")` and `(delz)/(delt)`
In the following, determine whether the following function is homogeneous or not. If it is so, find the degree.
f(x, y) = x2y + 6x3 + 7
In the following, determine whether the following function is homogeneous or not. If it is so, find the degree.
g(x, y, z) = `sqrt(3x^2+ 5y^2+z^2)/(4x + 7y)`
If v(x, y) = `log((x^2 + y^2)/(x + y))`, prove that `x (del"v")/(delx) + y (del"u")/(dely) = 1`
If w(x, y, z) = `log((5x^3y^4 + 7y^2xz^4 - 75y^3zz^4)/(x^2 + y^2))`, find `x (del"w")/(delx) + y (del"w")/(dely) + z (del"w")/(delz)`
Choose the correct alternative:
If v(x, y) = log(ex + ey), then `(del"v")/(delx) + (del"u")/(dely)` is equal to
Choose the correct alternative:
If w(x, y) = xy, x > 0, then `(del"w")/(delx)` is equal to
Choose the correct alternative:
f u(x, y) = x2 + 3xy + y – 2019, then `(delu)/(delx) "|"_(((4 , - 5)))` is equal to
Choose the correct alternative:
If w(x, y, z) = x2(y – z) + y2(z – x)+ z2(x – y) then `(del"w")/(delz) + (del"w")/(dely) + (del"w")/(delz)` is
Choose the correct alternative:
If f(x, y, z) = xy + yz + zx, then fx – fz is equal to
