Euler’s Theorem & Corollary’s Examples for Homogeneous Functions

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Euler’s Theorem & corollary’s examples for homogeneous functions :

Euler’s Theorem & corollary’s examples for homogeneous functions Calculus A.L.A. Mech. Div. - A C. K. Pithawala College of Engineering & Technology

GROUP – A7:

GROUP – A7 Name Roll Number Enrolment Number Patel Brijesh 122 160090119058 Desai Jainish 123 160090119011 Tankariya Hardik 126 160090119117 Pathak Aloknath 131 160090119085 Lalluwadia Yash 135 160090119039

Euler's Theorem for a Function of Two Variables:

Euler's Theorem for a Function of Two Variables Theorem – If u is a homogeneous function of two variables x and y of degree n then…. x + y = nu u x y

[Q.1] Find x + y where u = (8x² + y²) (log x – log y).:

[Q.1] Find x + y where u = (8x² + y²) (log x – log y). u = (8x² + y² ) ( log x – log y ) = ( 8x² + y² ) log Replacing x by xt and y by yt , u = t²(8x² + y²)log Hence , u is a homogeneous function of degree 2. By Euler’s Theorem, x + y = 2u = 2(8x² + y² )( log x – log y )

[Q.2] Find x + y where u = (x³ – 5y³)(log x – log y).:

[Q.2] Find x + y where u = (x³ – 5y³ )( log x – log y ). u = (x³ – 5y³ )( log x – log y) = (x³ – 5y³)log Replacing x by xt and y by yt , u = t³ (x³ – 5y³) log Hence, u is a homogeneous function of degree 3. By Euler’s Theorem, x + y = 3u = 3 (x³ – 5y³) (log x – log y )

[Q.3] Find x + y where u = 7x²y + 6xy²:

[Q.3] Find x + y where u = 7x²y + 6xy² u = 7x²y + 6xy² Replacing x by xt and y by yt , u = t³ (7x²y + 6xy²) Hence, u is a homogeneous function of degree 3. By Euler’s Theorem, x + y = 3u = 3 (7x²y + 6xy²)

2. Euler's Theorem for a Function of Three Variables:

2. Euler's Theorem for a Function of Three Variables Theorem – If u is a homogeneous function of three variables x, y, z of degree n then…. x + y + z = nu u x y z

[Q.4] Find x + y + z where u = 9x²y² + 12xy³+ 3z⁴:

[Q.4] Find x + y + z where u = 9x²y² + 12xy³+ 3z ⁴ u = 9x²y² + 12xy³+ 3z⁴ Replacing x by xt , y by yt , and z by zt , u = t⁴ (9x²y² + 12xy³+ 3z⁴) Hence, u is a homogeneous function of degree 4. By Euler’s Theorem, x + y + z = 4u

[Q .5] Find x + y + z where u = :

[Q .5] Find x + y + z where u = u = Replacing x by xt , y by yt , and z by zt , u = t⁶ Hence, u is a homogeneous function of degree 6. By Euler’s Theorem, x + y + z = 6u

[Q.6] Find x + y + z where u = x³ + 2y³ + 3z³ + 4xyz:

[Q.6] Find x + y + z where u = x³ + 2y³ + 3z³ + 4xyz u = x³ + 2y³ + 3z³ + 4xyz Replacing x by xt , y by yt , and z by zt , u = t³ (x³ + 2y³ + 3z³ + 4xyz) Hence, u is a homogeneous function of degree 3 . By Euler’s Theorem, x + y + z = 3u

3. Deduction from Euler's Theorem:

3 . Deduction from Euler's Theorem Corollary 1 – If u is a homogeneous function of two variables x, y of degree n then; x² + 2xy + y² = n(n - 1)u Corollary 2 – If z = f(u) is a homogeneous function of degree n in variables x and y of then; x + y = n

Note::

Note: If u is a homogeneous function of degree n in variables x, y and z then; x + y + z = n Corollary 3 – If z = f(u) is a homogeneous function of degree n in variables x and y of then; x² + 2xy + y² = g(u)[g'(u) – 1] where , g(u) = n

[Q.7] Find x² + 2xy + y² where u = log :

[Q.7] Find x² + 2xy + y² where u = log u = log Replacing x by xt and y by yt , u = t⁰ log Hence, u is a homogeneous function of degree 0. By Corollary 1, x² + 2xy + y² = 0(0-1)u = 0

[Q.8] Find x² + 2xy + y² where u = log :

[Q.8 ] Find x² + 2xy + y² where u = log u = log Replacing x by xt and y by yt , u = t⁰ log Hence, u is a homogeneous function of degree 0. By Corollary 1, x² + 2xy + y² = 0(0-1)u = 0

[Q.9] Find x + y where u = log x + log y.:

[Q.9] Find x + y where u = log x + log y . u = log x + log y = log xy Replacing x by xt and y by yt , u = log [t²( xy )] u is a nonhomogeneous function. But = xy is a homogeneous function of degree 2. Let f(u) = By Corollary 2, x + y = n = 2 = 2

[Q.10] Find x + y + z where u = log(x²+y²+z²).:

[Q.10] Find x + y + z where u = log(x²+y²+z²). u = log (x² + y² + z²) Replacing x by xt , y by yt , and z by zt , u = log [t²(x² + y² + z²)] u is a nonhomogeneous function. But = x² + y² + z² is a homogeneous function of degree 2. Let f(u) = By Corollary 2, x + y = n = 2 = 2

[Q.11] Find x + y + z where u=log:

[Q.11] Find x + y + z where u=log u = log Replacing x by xt , y by yt , and z by zt , u = log [t ] u is a nonhomogeneous function. But = is a homogeneous function of degree 1. By Corollary 2, x + y = n = 1 = 1

[Q.12] Find x + y + z where u = log(x²y+y²z+xz²).:

[Q.12] Find x + y + z where u = log(x²y+y²z+xz² ). u = log (x²y + y²z + xz² ) Replacing x by xt , y by yt , and z by zt , u = log [t³(x²y + y²z + xz²)] u is a nonhomogeneous function. But = x²y + y²z + xz² is a homogeneous function of degree 3. Let f(u) = By Corollary 2, x + y = n = 3 = 3

[Q.13] Find x² + 2xy + y² where u = log:

[Q.13] Find x² + 2xy + y² where u = log u = log Replacing x by xt and y by yt , u = log [t ] u is a nonhomogeneous function. But = is a homogeneous function of degree 1. Let f(u) = By Corollary 3, x² + 2xy + y² = g(u)[g'(u)–1] Where, g(u) = n = 1 = g'(u) = 0 Hence, x² + 2xy + y² = (0 – 1) = -

Thank you:

Thank you Reference:- Calculus Textbook 1 st Edition Dr. Sailesh S. Patel Atul Prakashan

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