Homogeneous First-Order Differential Equations
A first-order differential equation
F(x,y) dx + G(x,y) dy = 0
is homogeneous if
both F(x,y) and G(x,y) are
homogeneous functions of the same degree, where a homogeneous function
of degree n
is defined by:
f (sx,sy)=snf(x,y) for all x, y, and s
Example:
f (sx,sy)=snf(x,y) for all x, y, and s
Example:
Determine
if (xy) dx + (x2
+ y2) dy
= 0 is homogeneous.
Solution:
For F(x,y) = xy:
F(sx,sy) = (sx)(sy) = s2xy = s2F(x,y) → F(x,y) is homogeneous of degree 2
For G(x,y) = x2 + y2:
G(sx,sy) = (sx)2 + (sy)2 = s2x2 + s2y2 = s2(x2 + y2) = s2G(x,y) → G(x,y) is homogeneous of degree 2
∴ As both F(x,y) = xy and x2 + y2 are both homogeneous of the same degree, (xy) dx + (x2 + y2) dy = 0 is homogeneous
A method for solving homogeneous equations conerns the capability to transform them into separable differential equations using the substitution:
y = xv; (with dy = x dv + v dx)
F(sx,sy) = (sx)(sy) = s2xy = s2F(x,y) → F(x,y) is homogeneous of degree 2
For G(x,y) = x2 + y2:
G(sx,sy) = (sx)2 + (sy)2 = s2x2 + s2y2 = s2(x2 + y2) = s2G(x,y) → G(x,y) is homogeneous of degree 2
∴ As both F(x,y) = xy and x2 + y2 are both homogeneous of the same degree, (xy) dx + (x2 + y2) dy = 0 is homogeneous
A method for solving homogeneous equations conerns the capability to transform them into separable differential equations using the substitution:
y = xv; (with dy = x dv + v dx)