Homogeneous differential equation

A differential equation can be homogeneous in either of two respects: the coefficients of the differential terms in the first order case could be homogeneous functions of the variables, or for the linear case of any order there could be no constant term.

A first-order ordinary differential equation in the form:

is a homogeneous type if both functions M(x, y) and N(x, y) are homogeneous functions of the same degree n. That is, multiplying each variable by a parameter  ${\displaystyle \lambda }$, we find

Thus,

In the quotient   ${\displaystyle {\frac {M(tx,ty)}{N(tx,ty)}}={\frac {M(x,y)}{N(x,y)}}}$, we can let   ${\displaystyle t=1/x}$   to simplify this quotient to a function ${\displaystyle f}$ of the single variable ${\displaystyle y/x}$:

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