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Unformatted text preview: ix in the last expression rotates a vector in the plane clockwise by an angle . 2.4 Fundamental Solution Theorem
Theorem 2.3. Let be an matrix. Then the initial value problem ,
has the unique solution 
. Proof. First demonstrate that the solution works. The differential equation is satisfied: 12 .
The initial condition is also satisfied: . To show that the solution is unique, suppose
Then use the product rule to obtain is another solution of the initial value problem. ,
where we have also used the fact that and
by property (ii) in section 2.3. Moreover
solutions must satisfy the same initial condition. □ is a
since the two Example. Mass-spring system. Consider a spring of natural length L, mass m and Hooke‟s
law constant k on a frictionless table: The figure shows the spring extended a distance
of motion gives beyond its natural length. Newton‟s law .
This ODE is affine, but we can make it linear by substituting . Then get .
This can be further simplified by introducing the dimensionless time
. Write as a first order system
, , where . 13 Consider the initial value problem, using : .
. According to the classification of linear systems the
solution is a center. By the Fundamental Theorem the solution of the initial value problem is
Finally express the answer in terms of :
where .■ and Consider the result of Theorem 2.3 for a set of initial condition vectors
Put the initial condition vectors together to form a matrix
together to form
. We then have .
and the solutions Theorem 2.4. The matrix initial value problem
Has the unique solution . A matrix initial value problem is used in the beautiful proof of the law of exponents for the
matrix exponential given by problem 6, parts (b) and (c)!
The fundamental matrix solution is the solution of the initial value problem
Its name arises from the fact that
solve the initial value problem  for any .
Check by calculating (t) and
using this formula. When is a constant matrix,
. We will return to fundamental matrix solution in section 2.8 when we will consider a
function of time....
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- Fall '14