ODE - Seconed Semester - LEC 3 Homogeneous Equations

Homogeneous Equations
A homogeneous equation can be transformed into a separable equation by a change of
variables.


Definition: An equation in differential form M(x,y) dx + N(x,y) dy = 0 is said to be
homogeneous, if when written in derivative form dy/dx = f( x,y ) = g(y/x)

We can use another approach to define a homogeneous equation.
Definition: A function F(x,y) of the variables x and y is called homogeneous of degree n
if for any parameter t
F(tx, ty) = tn F(x,y)
Example:
Given F(x,y) = x3 – 4x2y + y3, it is a homogeneous function of degree 3 since
F(tx,ty) = (tx)3 – 4(tx)2(ty) + (ty)3 = t3(x3 – 4x2y + y3).
Theorem: The O.D.E. in differential form M(x,y) dx + N(x,y) = 0 is a homogeneous
O.D.E. if M(x,y) and N(x,y) are homogeneous functions of the same degree.


Theorem: Given a homogeneous O.D.E., the change of variable y = vx transforms the
equation into a separable equation in the variables v and x.Theorem: Given a homogeneous O.D.E., the change of variable y = vx transforms the
equation into a separable equation in the variables v and x.Theorem: Given a homogeneous O.D.E., the change of variable y = vx transforms the
equation into a separable equation in the variables v and x.