If u =
3+y?
is homogeneous of then which of the following is true
Answers
Answer:
system of linear equations is said to be homogenous if sum of the powers of the variables in each term is same.
In other words we can say that if constant term is a zero in a system of linear equations.
Let's consider the system of linear homogeneous equations to be
a
1
x+b
1
y+c
1
z=0
a
2
x+b
2
y+c
2
z=0
a
3
x+b
3
y+c
3
z=0
By clear observation, x=0,y=0,z=0 is a solution of above system of equations.
This solution is known as trivial solution.
For non-trivial solution, consider first two equations from above system.
b
1
c
2
−c
1
b
2
x
=
c
1
a
2
−a
1
c
2
y
=
a
1
b
2
−b
1
a
2
z
=k
∴x=k(b
1
c
2
−c
1
b
2
),y=k(c
1
a
2
−a
1
c
2
),z=k(a
1
b
2
−b
1
a
2
)
Now putting these values in the third equation, we get,
k[a
3
(b
1
c
2
−c
1
b
2
)−b
3
(a
1
c
2
−c
1
a
2
)−c
3
(a
1
b
2
−b
1
a
2
)]=0
or, a
3
(b
1
c
2
−c
1
b
2
−b
3
)(a
1
c
2
−c
1
a
2
)−c
3
(a
1
b
2
−b
1
a
2
)=0 where [k
=0]
This is required condition for the above system of above homogeneous linear equations to have non-trivial solution.