Please complete page 271 #1,2,5,6,9,10,11,13,14,15 (from Braun4e287289)I would like thorough explanations/work shown for all the problems solved. If you have any questions, let me know.3.1 Algebraic properties of solutions of linear systems
choice of constants c
1
,c
2
, •••
,c
1
.
For example, consider the system of equa
tions
dx
( 0
dt
=
4
b
)x, x = (
~~).
(6)
This system of equations was derived from the secondorder scalar equa
tion
(d
2
y
I
dt
2
)+4y
=0 by setting x
1
=
y
and
x
2
=
dy
I
dt.
Since
y
1
(t)=cos2t
and
yit)
= sin2t are two solutions of the scalar equation, we know that
x(t) = ( x
1
(t)) = c
1
(
cos2t) + c
2
(
sin2t)
x
2
(t)
2sin2t
2cos2t
(
c
1
cos2t+c
2
sin2t)
=

2c
1
sin2t + 2c
2
cos2t
is a solution of (6) for any choice of constants c
1
and c
2
•
The next step in our gameplan is to show that every solution of (4) can
be expressed as a linear combination of finitely many solutions. Equiv
alently, we seek to determine how many solutions we must find before we
can generate all the solutions of (4). There is a branch of mathematics
known as linear algebra, which addresses itself to exactly this question, and
it is to this area that we now turn our attention.
EXERCISES
In each of Exercises 13 convert the given differential equation for the sin
gle variable
y
into a system of firstorder equations.
1.
d3y+(dy)2=0
2.
d3y+cosy=e'
3.
d4y+d2y=l
dt
3
dt
dt
3
dt
4
dt
2
4. Convert the pair of secondorder equations
d
2
y
dz
d
2
dy
3
2 0
~
+
3
+
2z
=
0
dt2
+
dt
+
y
=
'
dt2
dt
into a system of 4 firstorder equations for the variables
xl=y,
5.
(a) Lety(t) be a solution of the equationy"+y'+y=O. Show that
x(t)=(y(t))
y'(t)
is a solution of the system of equations
x=(
~
Dx.
271
Answer