Lecture XV
Review of Dynamic Mathematics II
I.
Examples
A.
Homogeneous First Order Linear Differential Equations
1.
Example 1:
x
x
+
=
3
0
Proposed solution
x t
Ae
t
( )
=

3
Given an initial value of x(0) = 5, what is the value of A?
Check the
solution:
(
29
x t
e
x
t
e
x
t
x t
e
e
t
t
t
t
( )
’( )
’( )
( )
=
= 
+
= 
+
=




5
15
3
15
3 5
0
3
3
3
3
2.
Example 2:
x
x
+
=
6
0
Proposed solution
x t
Ae
t
( )
=

6
Given the initial value of x(0)=4, what is the value of A?
Check the
solution:
(
29
x t
e
x
t
e
x
t
x t
e
e
t
t
t
t
( )
’( )
’( )
( )
=
= 
+
= 
+




4
24
24
6 4
6
6
6
6
B.
Non Homogeneous First Order Linear Differential Equations with a Constant
Right Hand Side
1.
Example 1:
x
x
+
=
3
6
Proposed solution
x t
x
e
t
( )
=

+

0
3
6
3
6
3
What is the particular solution to this problem?
What is the
complementary solution to this problem?
Given an initial condition of
x(0)=5, check the solution:
(
29
(
29
x t
e
x t
e
x
t
e
x
t
x t
e
e
t
t
t
t
t
( )
( )
’( )
’( )
( )
=

+
=
+
= 
+
= 
+
+
=





5
2
2
3
2
9
3
9
3 3
2
6
3
3
3
3
3
2.
Could you solve for the initial conditions, if you where given x(2) = 10?
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(
29
x
x
e
x
e
( )
(
)
2
2
2
10
10
2
2
0
4
0
4
=

+
=
=

+

3.
Example 2:
x
x
+
=
6
6
Proposed solution
x t
x
e
t
( )
=

+

0
6
6
6
6
6
Taking the initial solution of x(0)=4 and checking the solution:
(
29
x t
e
x
t
e
x
t
x t
e
e
t
t
t
t
( )
’( )
’( )
( )
=
+
= 
+
= 
+
+
=




3
1
18
6
18
6 3
1
6
6
6
6
6
C.
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 Fall '08
 Moss
 Differential Equations, Derivative, Linear Differential Equations, Order Linear Diﬀerential

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