Page 671 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition
P. 671
Show that every solution of has the form .
5.
Hint Let be a solution of the equation, and show that is constant.
Show that if A is diagonalizable and
6.
satisfies , then each is a linear combination of , , …, where , , …, are the eigenvalues of
A.
It is possible to solve a single differential equation by expressing the equation as a system and then using the method of
7. this section. For the differential equation , show that the substitutions and lead to the
system
Solve this system and then solve the original differential equation.
Use the procedure in Exercise 7 to solve .
8.
Discuss: How can the procedure in Exercise 7 be used to solve ? Carry out your ideas.
9.
10.
(a) By rewriting 8 in matrix form, show that the solution of the system in Example 2 can
be expressed as
This is called the general solution of the system.
(b) Note that in part (a), the vector in the first term is an eigenvector corresponding to the
eigenvalue and the vector in the second term is an eigenvector corresponding to
the eigenvalue . This is a special case of the following general result:
THEOREM
