## 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)=s

Example:

f (sx,sy)=s

^{n}f(x,y) for all x, y, and sExample:

Determine
if (xy) dx + (x

^{2}+ y^{2}) dy = 0 is homogeneous.Solution:

For F(x,y) = xy:

F(sx,sy) = (sx)(sy) = s

For G(x,y) = x

G(sx,sy) = (sx)

∴ As both F(x,y) = xy and x

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) = s

^{2}xy = s^{2}F(x,y) → F(x,y) is homogeneous of degree 2For G(x,y) = x

^{2}+ y^{2}:G(sx,sy) = (sx)

^{2}+ (sy)^{2}= s^{2}x^{2}+ s^{2}y^{2}= s^{2}(x^{2}+ y^{2}) = s^{2}G(x,y) → G(x,y) is homogeneous of degree 2∴ As both F(x,y) = xy and x

^{2}+ y^{2}are both homogeneous of the same degree, (xy) dx + (x^{2}+ y^{2}) dy = 0 is homogeneousA 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)