Homogeneous Function and Euler's Theorem

A function f(x,y) is said to be homogeneous function in which the power of each term is same.

A function f(x,y) is a homogeneous function of order n, if the degree of each of its terms in x and y is equal to n. Thus

a0 xn + a1xn-1y + a2xn-2y2 + ... + an-1xyn-1 + anyn is a homogeneous function of order n.

xn[a0 + a1(y/x) + a2(y/x)2 + ... + an-1(y/x)n-1 + an(y/x)n] = xnφ(y/x)

Euler's Theorem for homogeneous Equations


  • If z is a homogeneous function of x,y of order n, then
    x ∂z/∂x + y ∂z/∂y = nz

  • Deductions from Euler's equation
    1. If z is a homogeneous function of x,y of order n, and z = f(u), then
      x ∂u/∂x + y ∂u/∂y = nf(u)/f '(u)

    2. If z is a homogeneous function of x,y of order n, and z = f(u), then
      x22u/∂x2 + 2xy ∂2u/∂x∂y + y22u/∂y2 = g(u)[g'(u) - 1]
      where g(u) = n f(u)/f'(u)


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