subject: Introduction On Homogenous Equation: [print this page] The equation is said to homogeneous equation, when the function is homogeneous. It is written by keeping zero on one side of equal sign and the homogeneous function of all the variables on other side of equal sign.
The standard form for the homogeneous equation is
F(x) =0
Conditions for Homogeneous Equation
Homogeneous equation cannot be an inconsistent since the origin of the homogeneous equation is always the solution.
When we add the solution of homogeneous equation to the solution of non homogeneous equation, the resultant will be the solution of the same non homogeneous equation.
Considering homogeneous equations the following implications needed to consider: there are no constant terms.
1. The determinant of the homogeneous system is non-zero where we have n variables and n equations, so the matrix corresponding is non- singular and the only solution to the equation is the vertex that is the origin. It is called the trivial solution to homogeneous equation.
2. If we have a lesser linearly independent homogeneous equations than the variables. Other is non trivial solutions to the equations, it can found by row reduction, with basic variables that is parametric form. Row reduction is the easy method because no need to handle the right side of the equation
Examples on Homogeneous Equations
Ex 1: (x2+3y2) dx - xy dy =0
Sol:dy/dx= '(x^2+3y^2) /(xy).'
= '(x / y) ' +3'(y/x)'
= '(y/x)'-1 + 3'(y/x)'
= g'(y/x)'
It is a homogeneous function
Ex 2: (x2+8y2) dx - xy dy =0
Sol:
dy/dx= '(x^2+8y^2) /(xy).'
= '(x/y) ' +8'(yx)'
= '(yx)' -1 + 8'(yx)'
= g'(y/x)'
It is a homogeneous function
Ex 3: Given F(x,y)= x3-5x2y+y3
Sol: It is a homogeneous function of degree 3
(tx,ty)= (tx)3-5(tx)2 (ty) + (ty)3
= t3(x3-5x2y+y3)
F(x,y)=F(tx,ty)
Ex 4: Given F(x,y)= x3-5x2y-3y3
Sol: It is a homogeneous function of degree 3
(tx,ty)= (tx)2-5(tx)2 (ty) - 3(ty)2
= t3(x3-5x2y-3y3 )
F(x,y)=F(tx,ty)
Ex 5: Given F(x,y)= x4-5x3y+y4
Sol: It is a homogeneous function of degree 4
(tx,ty)= (tx)4-5(tx)3 (ty) + (ty)4
= t4(x4-5x3y+y4 )
F(x, y) =F (tx, ty)
Ex 6: F(x,y)= 8x4+ 4xy +5y4 Find the equation is homogeneous or not?
