Math 150 (Lecture 22)
Repeated Eigenvalues
Consider the homogeneous system
x = Ax
where A has repeated eigenvalues i.e., det(A rI ) = 0 has a double root. It is possible
that we cannot nd two linearly independent eigenvectors to form a set of fundamental
Math 150 (Lecture 21)
Complex Eigenvalues
Consider the homogeneous system
x = Ax
where A is a constant 2 2 matrix. So x = ert is a solution if r is an eigenvalues and
a corresponding eigenvector of A. Recall that eigenvalues r1 and r2 of A are roots of t
Math 150 (Lecture 19)
Systems of Linear Algebraic Equations
Denition 1. Systems of n linear algebraic equations with n variables:
a11 x1 + a12 x2 + + a1n xn
= b1
.
.
.
an1 x1 + an2 x2 + + ann xn = bn
It can be written as Ax = b where
a11
.
A= .
.
an1
a12
Math 150 (Lecture 18)
Systems of First Order Linear Equations
Denition 1. A general system of rst order ODE:
x1 = F1 (t, x1 , x2 , . . . , xn )
x2 = F2 (t, x1 , x2 , . . . , xn )
.
.
.
xn = Fn (t, x1 , x2 , . . . , xn )
The system is said to have a soluti
Math 150 (Lecture 20)
Basic Theory of Systems of First Order Linear ODE
Consider the initial value problem for systems of rst order linear ODE in matrix form:
x = P (t)x + g (t), x(t0 ) = x0
x0
1
.
Notice that x0 = . is a constant column vector.
.
x0
n
Th
The God called Poetry
by Robert Graves (1895-1985)
Poetry as external, spiritual inspiration, a
transcendental force prior to nature; omnipotent but
unpredictable: traditional belief, poetry and religion,
the poet as prophet; poetry as didactic and offer
9/8/2015
Sonnet: On Being Cautioned Against Walking on an Headland Overlooking the Sea, Because It Was Frequented by a Lunatic by Charlotte Smith : The Poetry F
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