Homogeneous differential equation

A differential equation can be homogeneous in either of two respects.

A first order differential equation is said to be homogeneous if it may be written

${\displaystyle f(x,y)\,dy=g(x,y)\,dx,}$

where f and g are homogeneous functions of the same degree of x and y.^[1] In this case, the change of variable y = ux leads to an equation of the form

${\displaystyle {\frac {dx}{x}}=h(u)\,du,}$

which is easy to solve by integration of the two members.

Otherwise, a differential equation is homogeneous if it is a homogeneous function of the unknown function and its derivatives. In the case of linear differential equations, this means that there are no constant terms. The solutions of any linear ordinary differential equation of any order may be deduced by integration from the solution of the homogeneous equation obtained by removing the constant term.