Total solutions and Problems Part-04
Unit-05
Recurrence Relation:
Definition
Examples (Fibonacci,
Factorial etc.),
Linear recurrence
relations with constants coefficients –
Homogenous solutions
Particular solutions
Total solutions
Problems.
Total solutions-
1. The
total solution of a linear recurrence relation with constant coefficients is
the sum of the homogenous solution and the particular solution.
2. The
homogenous solution represents the general form of the sequence that satisfies
the recurrence relation without considering the initial conditions, while the
particular solution represents a particular sequence that satisfies both the
recurrence relation and the initial conditions.
3. To
find the total solution, we first find the homogenous solution by solving the
characteristic equation of the recurrence relation.
4. If
the roots of the characteristic equation are distinct, then the homogenous
solution has the following form:
a(n)
= A1r1^n + A2r2^n + ... + Ak*rk^n
where r1, r2, ..., rk
are the roots of the characteristic equation, and A1, A2, ..., Ak are
constants to be determined using the initial conditions of the sequence.
5. If
the roots of the characteristic equation are not distinct, then the homogenous
solution has additional terms that depend on the multiplicity of the roots.
6. Next,
we find the particular solution using the method of undetermined coefficients.
7. The
particular solution represents a particular sequence that satisfies both the
recurrence relation and the initial conditions.
8. Finally,
the total solution is the sum of the homogenous solution and the particular
solution.
9. The
constants in the homogenous solution are determined using the initial
conditions, while the constant in the particular solution is determined by
solving for it using the recurrence relation and the initial conditions.
10.
For example,
consider the linear recurrence relation:
a(n)
= 3a(n-1) - 2a(n-2) + 2^n
The characteristic
equation of this recurrence relation is:
r^2
- 3r + 2 = 0
which has roots 1 and 2.
Therefore, the homogenous
solution is:
a(n)
= A11^n + A22^n
where A1 and A2 are
constants to be determined using the initial conditions.
To find the particular
solution, we assume that it has the form:
a(n)
= C*2^n
where C is a constant to
be determined. Substituting this form into the recurrence relation, we obtain:
C2^n
= 3C2^(n-1) - 2C*2^(n-2) + 2^n
Simplifying and dividing
by 2^n, we obtain:
C
= 2
Therefore, the particular
solution is:
a(n)
= 2*2^n
The total solution
is the sum of the homogenous solution and the particular solution:
a(n)
= A11^n + A22^n + 2*2^n
where A1 and A2 are constants
determined by the initial conditions.
Problems-
1.
Find the 10th term of the sequence
defined by the linear recurrence relation a(n) = 2a(n-1) + 3a(n-2) with initial
conditions a(0) = 1 and a(1) = 2.
To
solve this problem, we first find the characteristic equation of the recurrence
relation:
r^2 - 2r - 3 = 0
which
has roots r = -1 and r = 3.
Therefore,
the homogenous solution is:
a(n) = A1*(-1)^n + A2*3^n
where
A1 and A2 are constants determined by the initial conditions.
Substituting the initial conditions, we obtain:
A1 + A2 = 1
A1 + 3*A2 = 2
Solving
this system of equations, we obtain
A1 = 1 and A2 = 1/2.
Therefore,
the total solution is:
a(n) = 1*(-1)^n + (1/2)*3^n
The
10th term of the sequence is:
a(10) = 1*(-1)^10 + (1/2)*3^10 = 5900.5
2.
Find the general form of the sequence
defined by the linear recurrence relation a(n) = 4a(n-1) - 4a(n-2) + 2^n with
initial conditions a(0) = 1 and a(1) = 3.
To
solve this problem, we first find the characteristic equation of the recurrence
relation:
r^2 - 4r + 4 = 0
which
has a repeated root r = 2.
Therefore,
the homogenous solution is:
a(n) = (A1 + A2*n)*2^n
where
A1 and A2 are constants determined by the initial conditions.
Substituting
the initial conditions, we obtain:
A1 = 1 A1 + 2*A2 = 3
Solving
this system of equations, we obtain
A1 = 1 and A2 = 1/2.
Therefore,
the general form of the sequence is:
a(n) = (1 + n/2)*2^n
3.
Find the particular solution of the
linear recurrence relation a(n) = 3a(n-1) - 2a(n-2) + 2^n with initial
conditions a(0) = 1 and a(1) = 2.
To
solve this problem, we use the method of undetermined coefficients to find the
particular solution.
We
assume that the particular solution has the form:
a(n) = C*2^n
Substituting
this form into the recurrence relation, we obtain:
C2^n = 3C2^(n-1) - 2C*2^(n-2) + 2^n
Simplifying
and dividing by 2^n, we obtain: C = 2
Therefore,
the particular solution is:
a(n) = 2*2^n
The
total solution is the sum of the homogenous solution and the particular
solution, which we have already found in previous examples.
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