1. Homogeneous Functions; Equations with Homogeneous Coefficients
MATH 374 Lecture 5
Homogeneous Functions; Equations with Homogeneous Coefficients

2. 2.2: Homogeneous Functions
Definition: A function f(x,y) is said to be homogeneous of degree k (in x and y) if and only if f( x, y) = k f(x,y).
Example 1: Show is homogeneous of degree ½.
Solution: 2

3. Example 2
Let M(x,y) and N(x,y) be homogeneous of degree k. Is M(x,y)/N(x,y) homogeneous? If so, of what degree?
Solution:
This implies that M/N is homogeneous of degree zero.
This example proves a theorem! 3

4. Properties of Homogeneous Functions
Theorem 2.1: If M(x,y) and N(x,y) are both homogeneous of the same degree k, then the function M(x,y)/N(x,y) is homogeneous of degree zero.
A useful property of homogeneous functions is given by the next theorem.
Theorem 2.2: If f(x,y) is homogeneous of degree zero in x and y, then f(x,y) is a function of y/x alone.

5. Proof of Theorem 2.2 Proof: Let y = vx.
Then f(x,y) = f(x,vx) = x0f(1,v), since f is homogeneous of degree zero.
Hence f(x,y) = f(1,y/x), i.e. a function of y/x.

6. 2.3: Equations with Homogeneous Coefficients
Suppose we can write an equation of order one in the form
M(x,y) dx + N(x,y) dy = 0, (1)
with M and N homogeneous, of the same degree in x and y.
Then Theorem 2.1 tells us M/N is homogeneous of degree zero and Theorem 2.2 says that M/N is a function of y/x alone.
Therefore, (1) can be put in the form
dy/dx + g(y/x) = 0. (2)
If we let y = vx, then dy/dx = dv/dx·x + v·1, so (2) can be written
x dv/dx + v + g(v) = 0, (3)
which is separable!
Solving (3) for v and letting v = y/x leads to a solution of (1).
Note: Letting x = vy to get an equation in y and v will also work. In this case, dx/dy = dv/dy·y + v.

7. Example 1
Solve
Solution: Note that (4) is not separable, but it can be written in the form:
(x-3y)dx + (-3x-y)dy = 0,
i.e. in the form M(x,y)dx + N(x,y)dy=0, with
M(x,y) = x-3y and N(x,y) = -3x-y.
M and N are homogeneous of degree 1, so we can let y = vx to rewrite (4) as a separable equation.
From (4), we see that

8. Example 1 (continued)
Equation (5) is separable:

9. Example 1 (continued)

10. Example 1 (continued)

11. Example 1 (continued)