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

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

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

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.

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.

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.

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

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

Example 1 (continued)

Example 1 (continued)

Example 1 (continued)