Introduction
Applications of Differential Equations
Checking Solutions and IVP
Evaporation Example
Nonautonomous Example
Introduction to Maple
Introduction to Maple
1
Introduction to Maple
: A Symbolic Math Program
We enter a function
y
(
t
) = 3
e

t
cos(2
t
),
y
:=
t
→
3
·
exp(

t
)
·
cos(2
·
t
);
The arrow is

and
>
and multiplication is
*
. To plot this function
plot
(
y
(
t
)
, t
= 0
..
2
·
Pi);
Joseph M. Mahaffy,
h
[email protected]
i
Lecture Notes – Introduction to Differential Eq
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The Class — Overview
The Class...
Introduction
Applications of Differential Equations
Checking Solutions and IVP
Evaporation Example
Nonautonomous Example
Introduction to Maple
Introduction to Maple
2
We have the function:
y
(
t
) = 3
e

t
cos(2
t
),
This can be differentiated (and stored in variable
dy
) by typing
dy
:=
diff
(
y
(
t
)
, t
);
Maple gives:
dy
:=

3
e

t
cos(2
t
)

6
e

t
sin(2
t
)
The absolute minimum and a relative maximum are found with
Maple:
tmin
:=
fsolve
(
dy
= 0
, t
= 1
..
2);
y
(
tmin
);
tmax
:=
fsolve
(
dy
= 0
, t
= 2
.
5
..
3
.
5);
y
(
tmax
);
The result was an
absolute minimum
at (1
.
33897
,

0
.
703328).
The result was a
relative maximum
at (2
.
90977
,
0
.
1462075).
Joseph M. Mahaffy,
h
[email protected]
i
Lecture Notes – Introduction to Differential Eq
— (44/47)
The Class — Overview
The Class...
Introduction
Applications of Differential Equations
Checking Solutions and IVP
Evaporation Example
Nonautonomous Example
Introduction to Maple
Introduction to Maple
3
With
y
(
t
) = 3
e

t
cos(2
t
), we can solve
Z
3
e

t
cos(2
t
)
dt
and
Z
5
0
3
e

t
cos(2
t
)
dt
These can be integrated by typing
int
(
y
(
t
)
, t
);
int
(
y
(
t
)
, t
= 0
..
5);
evalf
(%);
For the indefinite integral, Maple gives:

3
5
e

t
cos(2
t
) +
6
5
e

t
sin(2
t
)
For the definite integral, Maple gives:
3
5

3
5
e

5
cos(10) +
6
5
e

5
sin(10)
=
0
.
59899347
Joseph M. Mahaffy,
h
[email protected]
i
Lecture Notes – Introduction to Differential Eq
— (45/47)
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The Class — Overview
The Class...
Introduction
Applications of Differential Equations
Checking Solutions and IVP
Evaporation Example
Nonautonomous Example
Introduction to Maple
Introduction to Maple
4
Show
y
(
t
) = 3
e

t
cos(2
t
) is a solution of the differential equation
y
00
+ 2
y
0
+ 5
y
= 0
.
The function and derivatives are entered by
y
:=
t
→
3
·
exp(

t
)
·
cos(2
·
t
);
dy
:=
diff
(
y
(
t
)
, t
);
sdy
:=
diff
(
y
(
t
)
, t
$2);
If we type
sdy
+ 2
·
dy
+ 5
·
y
(
t
);
Maple gives
0
, which verifies this is a solution.
Joseph M. Mahaffy,
h
[email protected]
i
Lecture Notes – Introduction to Differential Eq
— (46/47)
The Class — Overview
The Class...
Introduction
Applications of Differential Equations
Checking Solutions and IVP
Evaporation Example
Nonautonomous Example
Introduction to Maple
Introduction to Maple
5
Maple finds the general solution to the differential equation
de
:=
diff
(
Y
(
t
)
, t
$2) + 2
·
diff
(
Y
(
t
)
, t
) + 5
·
Y
(
t
) = 0;
dsolve
(
de, Y
(
t
));
Maple produces
Y
(
t
) =
C
1
e

t
sin(2
t
) +
C
2
e

t
cos(2
t
)
To solve an initial value problem, say
Y
(0) = 2 and
Y
0
(0) =

1, enter
dsolve
(
{
de, Y
(0) = 2
, D
(
Y
)(0) =

1
}
, Y
(
t
));
Maple produces
Y
(
t
) =
1
2
e

t
sin(2
t
) + 2
e

t
cos(2
t
)
,
which is made into a useable function by typing
Y
:=
unapply
(
rhs
(%)
, t
);
Joseph M. Mahaffy,
h
[email protected]
i
Lecture Notes – Introduction to Differential Eq
— (47/47)
 Fall '08
 staff