Differential Equations - Ordinary Differential Equations - Linear Systems Of Differential Equations
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Example 7.2. The male population x1 and female population x2 of a bird community have a constant death rate k and a variable birth rate r (t). This community may be modelled by
x˙1 = −kx1 + r (t)x2
and
x˙2 = −kx2 + r (t)x2 ,
that is, by the linear system
x˙ =
−k 0
r (t) x. −k + r (t)
⊠
In order to consider the general solution of a linear system of differential equations we shall first consider the homogeneous system
x˙ = P(t)x
(I.8)
and look at linear independence. OHP 31
Definition. The functions x(1) , . . . , x(m) are said to be linearly dependent on the interval (α, β) if there exist ci (not all zero) such that m X
ci x(i) (t) ≡ 0,
α < t < β.
(I.9)
i=1
They are said to be linearly independent if they are not linearly dependent. We can test for the linear independence of n solutions by calculating their Wronskian. Thus if x(1) , . . . , x(n) are n solutions of (I.8), then their Wronskian is given by W [x(1) , . . . , x(n) ] = det(x(1) , . . . , x(n) ), that is, the determinant of the matrix whose columns are the solutions x(1) , . . . , x(n) . The solutions are linearly independent if and only if the Wronskian is nonzero. Definition. Any set of solutions {x(1) , . . . , x(n) } of the linear homogeneous system (I.8) which is linearly independent at each point on the interval (α, β) is said to be a fundamental set of solutions. OHP 32
Definition. Suppose that x(1) , . . . , x(n) form a fundamental set of solutions for (I.8). Then the n × n matrix h i (1) (2) (n) Ψ(t) = x (t), x (t), . . . , x (t) ,
whose columns are the solutions x(1) , . . . x(n) is said to be a fundamental matrix for the homogeneous system ˙ (I.8). (It is then not hard to see that Ψ(t) = P(t)Ψ(t).) Theorem I.3. If x(1) , . . . , x(n) form a fundamental set of solutions for (I.8), then every solution of (I.8) may be expressed as x(t) = c1 x(1) (t) + · · · + cn x(n) (t),
t ∈ (α, β),
where c1 , . . . , cn are arbitrary constants. This may be written as x(t) = Ψ(t)c, where c is a n ×1 matrix having components c1 , . . . , cn . Proof. Omitted.
Example 7.3. It may be verified that two solutions of the system 1 −1 x˙ = x 2 4 OHP 33
are given by x(1) (t) =
e2t −e2t
and
x(2) (t) =
e3t −2e3t
.
A Wronskian calculation shows that it has the value −e5t 6= 0 and so these two solutions are linearly independent. Hence any solution of the system is given by x = c1 x(1) + c2 x(2) and a fundamental matrix is Ψ(t) =
e2t −e2t
e3t −2e3t
.
⊠
Example 7.4. It may be verified that the system
x˙ =
"
0 1 1 0 1 1
# 1 1 x 0
has solutions OHP 34
x˙1 = −kx1 + r (t)x2
and
x˙2 = −kx2 + r (t)x2 ,
that is, by the linear system
x˙ =
−k 0
r (t) x. −k + r (t)
⊠
In order to consider the general solution of a linear system of differential equations we shall first consider the homogeneous system
x˙ = P(t)x
(I.8)
and look at linear independence. OHP 31
Definition. The functions x(1) , . . . , x(m) are said to be linearly dependent on the interval (α, β) if there exist ci (not all zero) such that m X
ci x(i) (t) ≡ 0,
α < t < β.
(I.9)
i=1
They are said to be linearly independent if they are not linearly dependent. We can test for the linear independence of n solutions by calculating their Wronskian. Thus if x(1) , . . . , x(n) are n solutions of (I.8), then their Wronskian is given by W [x(1) , . . . , x(n) ] = det(x(1) , . . . , x(n) ), that is, the determinant of the matrix whose columns are the solutions x(1) , . . . , x(n) . The solutions are linearly independent if and only if the Wronskian is nonzero. Definition. Any set of solutions {x(1) , . . . , x(n) } of the linear homogeneous system (I.8) which is linearly independent at each point on the interval (α, β) is said to be a fundamental set of solutions. OHP 32
Definition. Suppose that x(1) , . . . , x(n) form a fundamental set of solutions for (I.8). Then the n × n matrix h i (1) (2) (n) Ψ(t) = x (t), x (t), . . . , x (t) ,
whose columns are the solutions x(1) , . . . x(n) is said to be a fundamental matrix for the homogeneous system ˙ (I.8). (It is then not hard to see that Ψ(t) = P(t)Ψ(t).) Theorem I.3. If x(1) , . . . , x(n) form a fundamental set of solutions for (I.8), then every solution of (I.8) may be expressed as x(t) = c1 x(1) (t) + · · · + cn x(n) (t),
t ∈ (α, β),
where c1 , . . . , cn are arbitrary constants. This may be written as x(t) = Ψ(t)c, where c is a n ×1 matrix having components c1 , . . . , cn . Proof. Omitted.
Example 7.3. It may be verified that two solutions of the system 1 −1 x˙ = x 2 4 OHP 33
are given by x(1) (t) =
e2t −e2t
and
x(2) (t) =
e3t −2e3t
.
A Wronskian calculation shows that it has the value −e5t 6= 0 and so these two solutions are linearly independent. Hence any solution of the system is given by x = c1 x(1) + c2 x(2) and a fundamental matrix is Ψ(t) =
e2t −e2t
e3t −2e3t
.
⊠
Example 7.4. It may be verified that the system
x˙ =
"
0 1 1 0 1 1
# 1 1 x 0
has solutions OHP 34
