5 2  Homogeneous differential equations
Consider the following functions in x and y
F1 (x, y) = y2 + 2xy,
F2 (x, y) = 2x – 3y,
F3 (x, y) = cos
y
x
⎛
⎞
⎜
⎟
⎝
⎠ ,
F4 (x, y) = sin x + cos y
If we replace x and y by λx and λy respectively in the above functions, for any nonzero
constant λ, we get
F1 (λx, λy) = λ2 (y2 + 2xy) = λ2 F1 (x, y)
F2 (λx, λy) = λ (2x – 3y) = λ F2 (x, y)
F3 (λx, λy) = cos
cos
y
y
x
x
⎛λ
⎞
⎛
⎞
=
⎜
⎟
⎜
⎟
⎝λ
⎠
⎝
⎠  = λ0  F3 (x, y)
F4 (λx, λy) = sin λx + cos λy ≠ λn F4 (x, y), for any n ∈ N
Here, we observe that the functions F1, F2, F3 can be written in the form
F(λx, λy) = λn F (x, y) but F4 can not be written in this form This leads to the following
definition:
A function F(x, y) is said to be homogeneous function of degree n if
F(λx, λy) = λn F(x, y) for any nonzero constant λ We note that in the above examples, F1, F2, F3 are homogeneous functions of
degree 2, 1, 0 respectively but F4 is not a homogeneous function