supper position (1)
DESCRIPTION
Supper PositionTRANSCRIPT
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Full-2008 Advanced Mathematics GE604 Ordinary Differential Equation
Page 1 of 38
Q1 Find the solution for ( ) ( ) ( )
With conditions ( ) ( ) by using :
1-Homogeneous and particular solution.
2-Using superposition.
3-Using Laplace transform.
Solution :-
1- Homogeneous and particular solution :
( ) {
Homogeneous solution
( ) ( )
-
Full-2008 Advanced Mathematics GE604 Ordinary Differential Equation
Page 2 of 38
particular solution
( ) ( )
( )
( ) ( )
( )
( )
( )
( ) ( )
Now repeat last steps for initial condition :
( )
( )
( )
( )
( )
Homogeneous solution
( ) ( )
particular solution
-
Full-2008 Advanced Mathematics GE604 Ordinary Differential Equation
Page 3 of 38
( ) ( )
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( )
( ) ( )
2-Using superposition :
( ) [ ( ) ( )] ( )
( ) ( ) ( )
=
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Full-2008 Advanced Mathematics GE604 Ordinary Differential Equation
Page 4 of 38
( )
( )
( )
( )
From equation (1) we get :
( ) ( )
( ) ( ) ( )
( ) ( ( ))
( )
3-Using Laplace transform:
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-
Full-2008 Advanced Mathematics GE604 Ordinary Differential Equation
Page 5 of 38
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Another method:
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Full-2008 Advanced Mathematics GE604 Ordinary Differential Equation
Page 6 of 38
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Full-2008 Advanced Mathematics GE604 Ordinary Differential Equation
Page 7 of 38
Q2 Find the solution for ( ) ( ) ( )
With conditions ( ) ( ) by using :
1-Homogeneous and particular solution.
2-Using superposition.
3-Using Laplace transform.
Solution :-
1- Homogeneous and particular solution :
( ) ( ) {
Homogeneous solution
( ) ( )
-
Full-2008 Advanced Mathematics GE604 Ordinary Differential Equation
Page 8 of 38
particular solution
( ) ( )
( )
( ) ( )
( ) ( ) ( )
( )
( ) ( ) ( )
( ) ( )
Now repeat last steps for initial condition :
( )
( )
( )
( )
( )
( )
Homogeneous solution
( ) ( )
particular solution
-
Full-2008 Advanced Mathematics GE604 Ordinary Differential Equation
Page 9 of 38
( ) ( )
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2-Using superposition :
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( ) ( ( ) )
( ) ( )
-
Full-2008 Advanced Mathematics GE604 Ordinary Differential Equation
Page 10 of 38
=
3-Using Laplace transform:
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-
Full-2008 Advanced Mathematics GE604 Ordinary Differential Equation
Page 11 of 38
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Full-2008 Advanced Mathematics GE604 Ordinary Differential Equation
Page 12 of 38
Another method:
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-
Full-2008 Advanced Mathematics GE604 Ordinary Differential Equation
Page 13 of 38
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Q3 Find the solution for ( ) ( ) ( )
With conditions ( ) ( ) by using :
1-Homogeneous and particular solution.
2-Using superposition.
3-Using Laplace transform.
Solution :-
-
Full-2008 Advanced Mathematics GE604 Ordinary Differential Equation
Page 14 of 38
1- Homogeneous and particular solution :
( ) {
Homogeneous solution
( ) ( )
particular solution
( ) ( )
( )
( ) ( )
( ) ( ) ( )
( )
( ) ( ) ( )
( ) ( )
Now repeat last steps for initial condition :
( )
( )
( )
( )
-
Full-2008 Advanced Mathematics GE604 Ordinary Differential Equation
Page 15 of 38
( )
Homogeneous solution
( ) ( )
particular solution
( ) ( )
( )
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( )
( )
( )
( ) ( )
2-Using superposition :
( ) ( ) ( ) ( )
( ) ( )
For
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Full-2008 Advanced Mathematics GE604 Ordinary Differential Equation
Page 16 of 38
=
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Full-2008 Advanced Mathematics GE604 Ordinary Differential Equation
Page 17 of 38
Q4 Find the solution for ( ) ( ) ( )
With conditions ( ) ( ) by using :
1-Homogeneous and particular solution.
2-Using superposition.
3-Using Laplace transform.
)(tf
t2/ 2/3
Solution :-
1- Homogeneous and particular solution :
( )
{
Homogeneous solution
( ) ( )
-
Full-2008 Advanced Mathematics GE604 Ordinary Differential Equation
Page 18 of 38
particular solution
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( )
( ) ( )
( )
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( )
( )
( )
Now repeat last steps for initial condition for
:
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Homogeneous solution
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particular solution
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-
Full-2008 Advanced Mathematics GE604 Ordinary Differential Equation
Page 19 of 38
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Now repeat last steps for initial condition for
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Homogeneous solution
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particular solution
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-
Full-2008 Advanced Mathematics GE604 Ordinary Differential Equation
Page 20 of 38
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2-Using superposition :
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-
Full-2008 Advanced Mathematics GE604 Ordinary Differential Equation
Page 21 of 38
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)(tf
t2/ 2/3 =
t
t
)(tf
2/
t
t
)(tf
2/ 2/3
+
t
t
)(tf
2/ 2/3
+
t
t
)(tf
2/ 2/3
3-Using Laplace transform:
-
Full-2008 Advanced Mathematics GE604 Ordinary Differential Equation
Page 22 of 38
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Full-2008 Advanced Mathematics GE604 Ordinary Differential Equation
Page 23 of 38
Q5 Find the solution for ( ) ( ) ( )
With conditions ( ) ( ) by using :
1-Homogeneous and particular solution.
2-Using superposition.
3-Using Laplace transform.
Solution :-
1- Homogeneous and particular solution :
( ) {
Homogeneous solution
( ) ( )
( )
-
Full-2008 Advanced Mathematics GE604 Ordinary Differential Equation
Page 24 of 38
particular solution
( ) ( )
( )
( ) ( )
( )
( )
( )
( ) ( )
Now repeat last steps for initial condition :
( )
( )
( )
( )
( )
Homogeneous solution
( ) ( )
particular solution
( ) ( )
-
Full-2008 Advanced Mathematics GE604 Ordinary Differential Equation
Page 25 of 38
( )
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2-Using superposition :
( ) ( ) ( )
From equation (1) we get :
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( )
-
Full-2008 Advanced Mathematics GE604 Ordinary Differential Equation
Page 26 of 38
=
3-Using Laplace transform:
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-
Full-2008 Advanced Mathematics GE604 Ordinary Differential Equation
Page 27 of 38
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-
Full-2008 Advanced Mathematics GE604 Ordinary Differential Equation
Page 28 of 38
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Q6 Find the solution for ( ) ( ) ( )
With conditions ( ) ( ) by using :
1-Homogeneous and particular solution.
2-Using superposition.
3-Using Laplace transform.
Solution :-
-
Full-2008 Advanced Mathematics GE604 Ordinary Differential Equation
Page 29 of 38
1- Homogeneous and particular solution :
( ) {
Homogeneous solution
( ) ( )
particular solution
( ) ( )
( )
( ) ( )
( )
( )
( )
( ) ( )
Now repeat last steps for initial condition :
( )
( )
( )
( )
-
Full-2008 Advanced Mathematics GE604 Ordinary Differential Equation
Page 30 of 38
( )
Homogeneous solution
( ) ( )
particular solution
( ) ( )
( )
( ) ( )
( )
( )
( )
( ) ( )
( )
( )
( )
( )
( )
( )
Homogeneous solution
( ) ( )
particular solution
-
Full-2008 Advanced Mathematics GE604 Ordinary Differential Equation
Page 31 of 38
( ) ( )
( ) ( )
( )
( )
( )
( ) ( )
2-Using superposition :
( ) ( ) ( ) ( ) ( )
( )
( )
( )
( )
( )
-
Full-2008 Advanced Mathematics GE604 Ordinary Differential Equation
Page 32 of 38
=
3-Using Laplace transform:
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|
(
) |
-
Full-2008 Advanced Mathematics GE604 Ordinary Differential Equation
Page 33 of 38
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-
Full-2008 Advanced Mathematics GE604 Ordinary Differential Equation
Page 34 of 38
Q7 Find the solution for ( ) ( ) ( ) Extension: Eng.ALI A.IHBEEL
With conditions ( ) ( ) by using :
1-Homogeneous and particular solution.
2-Using superposition.
3-Using Laplace transform.
t
F(t)
5.0
Solution:
1- Homogeneous and particular solution:
( ) {
}
Homogeneous solution:
( ) ( )
-
Full-2008 Advanced Mathematics GE604 Ordinary Differential Equation
Page 35 of 38
particular solution:
( ) ( )
( )
( ) ( )
( ) ( )
( )
( )
( )
( )
Now repeat last steps for initial conditions for
:
( )
(
) (
) (
) (
)
( )
(
) (
) (
)
Homogeneous solution:
( ) ( )
( )
( ) ( )
(
)
-
Full-2008 Advanced Mathematics GE604 Ordinary Differential Equation
Page 36 of 38
( )
(
)
( ) ( )
( )
2-Using superposition :
( ) ( ) (
)
( ) ( ) (
)
( )
( ) [ (
) (
) (
)]
( )
( ) ( )
t
F(t)
5.0
5.0t
F(t)
-
Full-2008 Advanced Mathematics GE604 Ordinary Differential Equation
Page 37 of 38
3-Using Laplace transform:
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}
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)
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)
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)
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-
Full-2008 Advanced Mathematics GE604 Ordinary Differential Equation
Page 38 of 38
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