Question

What is the meaning of homogeneous differential equation?

Answer 1

A differential equation is an algebraic equation in unknown variables like x y z etc. that includes variables, powers of variables, derivatives of variables and various powers of the derivatives. Often some initial conditions are also given with some values for the variables and the derivatives.

The solution of the differential equation is a relation among the variables using algebraic functions.

A homogeneous differential equation consists of algebraic terms of equal degree. The sum of exponents of the variables or their derivatives in each term is same.

if the sums of exponents in all terms aren't equal then the differential equation is not homogeneous.

Answer 2

Answer:

Step-by-step explanation:

In first-order ODEs, we say that a differential equation in the form

dydx=f(x,y)

is said to be homogeneous if the function f(x,y) can be expressed in the form f(yx), and then solved by the substitution z=yx.

In second-order ODEs, we say that a differential equation in the form

ad2ydx2+bdydx+cy=f(x)

is said to be homogeneous if f(x)=0.

Is there a relation between these two? What does homogeneous mean? I thought it's when something =0, because in linear algebra, a system of n equations is homogeneous if it is in the form Ax=0n×1; but this doesn't seem to be the case for first-order ODEs.