(1)
a) Verify that each of the following are solutions of (1):
and 
b) Let a and b be arbitrary constants. Verify that
is
a solution of (1).
c) Verify that
is the solution of (1) with initial conditions 
.
Readings Questions:
You should read the section before attempting any reading questions or problems. Remember that you are responsible for all definitions and examples. Also remember to keep the answers to the reading questions on a separate piece of paper; these will be randomly collected from time to time.
1) Define the following:
a) linear nth order linear differential equation
b) homogeneous differential equation
c) characteristic equation
2) Consider the differential equation
on page 275:
(1)
a) Verify that each of the following are solutions of (1):
and
b) Let a and b be arbitrary constants. Verify that is
a solution of (1).
c) Verify that
is the solution of (1) with initial conditions .
3) Compare theorem 2 on page 276 to theorem 1 on page 23.
4) Determine if the following pairs
of functions are linearly independent over the real line.
a) 
b) 
5) Let 
be an arbitrary function. Show that the zero function and
are linearly dependent.
6) On page 279 the book says that
if
and 
are linearly dependent then 
.
I want you to think about the other direction: if
and 
for each x then
and
are linearly dependent.
7) Explain why a second order linear
differential equation will always have two linearly
independent solutions.
8) Can the differential equation 
have any singular solutions? Explain. (Note that each
of 
are given constants.)
9) Consider Eq(18) on page 282. Explain
why the hypothesis that r1 and r2 are distinct roots
of (18) implies 
are linearly independent.
10) Verify that 
are linear independent solutions of 
.
11) Verify that 
are linear independent solutions of 
.
12) Compare the following:
a) theorem 1 on page 287 and page 274
b) theorem 2 on page 288 and page 276
c) definition of linearly dependent functions on page 290 and page 278
d) Wronskian on page 291and page 279
e) theorem 3 on page 294 and page 279
f) theorem 4 on page 295 and page 280
Problems: page 284 5, 12, 16, 17, 20, 23, 29, 36, 41, 46, 48, 49
page 297 4, 7, 11, 13, 18, 21, 24, 26, 27, 30