Math A285
Di↵erential Equations §2.5: Solutions by Substitutions
Instructor: Scott Northrup
, Homogenous Equations
If a function f possesses the property f (tx, ty ) = t ↵ f (x, y ) for
some real number ↵, then f is said to be a homogenous function
of degree ↵. For example, f (x, y ) = x 3 + y 3 is a homogeneous
function of degree 3, since
f (tx, ty ) = (tx)3 + (ty )3 = t 3 (x 3 + y 3 ) = t 3 f (x, y )
A first order di↵erential equation in di↵erential form
M(x, y ) dx + N(x, y ) dy = 0
is said to be homogeneous* if both the coefficients M and N are
homogenous functions of the same degree.
Di↵erential Equations §2.5: Solutions by Substitutions
Instructor: Scott Northrup
, Homogenous Equations
If a function f possesses the property f (tx, ty ) = t ↵ f (x, y ) for
some real number ↵, then f is said to be a homogenous function
of degree ↵. For example, f (x, y ) = x 3 + y 3 is a homogeneous
function of degree 3, since
f (tx, ty ) = (tx)3 + (ty )3 = t 3 (x 3 + y 3 ) = t 3 f (x, y )
A first order di↵erential equation in di↵erential form
M(x, y ) dx + N(x, y ) dy = 0
is said to be homogeneous* if both the coefficients M and N are
homogenous functions of the same degree.
