homogeneous equation [help]

define homogeneous equation

hi
pls define the homogeneous equation for me.

Any equation or system of equations that satisfies

Ax = 0.

An inhomogeneous system can be written on the form

Ax = b.

Thus a homogeneous equation is a special case of the inhomogeneous equations where the constants are set to zero. Ax can be a polynomial, contain derivatives or integrals or any other operator.

hi Jone
can it be a matrix?

Yes, it can be a matrix equation which is the same as a system of equations. For example

ax1 + bx2 = 0
cx1 + dx2 = 0

which can be written more compactly as

Ax = 0, where A = [a b; c d] and x = [x1; x2].

There are two definitions of the term “homogeneous differential equation.” One definition calls a first-order equation of the form:

M(x,y)+N(x,y)=0

homogeneous if M and N are both homogeneous functions of the same degree. The second definition — and the one which you'll see much more often—states that a differential equation (of any order) is homogeneous if once all the terms involving the unknown function are collected together on one side of the equation, the other side is identically zero.

For example,

Y''-2Y'+Y=0 is homogeneous
but
Y''-2Y'+Y=x is not.
The nonhomogeneous equation
a(x)y''+b(x)y'+c(x)y=d(x)...............(1)
can be turned into a homogeneous one simply by replacing the right-hand side by 0:
a(x)y''+b(x)y'+c(x)y=0...............(2)

Equation (2) is called the homogeneous equation corresponding to the nonhomogeneous equation, (1). There is an important connection between the solution of a nonhomogeneous linear equation and the solution of its corresponding homogeneous equation.The two principal results of this relationship are as follows:

Theorem A. If y1( x) and y2( x) are linearly independent solutions of the linear homogeneous equation (**), then every solution is a linear combination of y1 and y2. That is, the general solution of the linear homogeneous equation is

y=c1y1+c2y2

Theorem B. If y( x) is any particular solution of the linear nonhomogeneous equation (*), and if y h ( x) is the general solution of the corresponding homogeneous equation, then the general solution of the linear nonhomogeneous equation is

Y=Yh+Y'
That is,

General Solution(GS) of linear nonhomogeneous eqn=GS of corresopnding homogeneous eqn+particular solution of given nonhomogeneous eqn

[Note: The general solution of the corresponding homogeneous equation, which has been denoted here by Yh , is sometimes called the complementary function of the nonhomogeneous equation (1).] Theorem A can be generalized to homogeneous linear equations of any order, while Theorem B as written holds true for linear equations of any order. Theorems A and B are perhaps the most important theoretical facts about linear differential equations—definitely worth memorizing.



Homogeneous and Non-Homogeneous equations are Matrix equations.