Accumulation
Authors: Bill Davis, Horacio Porta and Jerry Uhl
©19962007
Publisher: Math Everywhere, Inc.
Version 6.0
2.06 More Tools and Measurements:
Techniques for Calculating Integrals
BASICS
B.1) Separating the variables and integrating to get formulas for solutions
of some differential equations
·
B.1.a)
Here is
Mathematica
's formula for the solution of the differential equation
y'
@
x
D
=
x
H
y
@
x
D
+
4
L
2
with y
@
3
D
= 
9.
Clear
@
y, x, sol
D
;
dsol
=
DSolve
A9
y'
@
x
D
==
x
H
y
@
x
D
+
4
L
2
,
y
@
3
D
== 
9
=
, y
@
x
D
, x
E
::
y
@
x
D
Ø 
2
I

81
+
10 x
2
M

43
+
5 x
2
>>
Use the technique of separating and integrating to explain where this formula comes from.
·
Answer:
Rewrite
y'
@
x
D
=
x
H
y
@
x
D
+
4
L
2
to get
y'
@
x
D
H
y
@
x
D
+
4
L
2
=
x.
Some folks call this technique of putting all the
y
@
x
D
terms on one side
and the
x
terms on the other side by the name
"separation of variables."
Integrate both sides from
3 to
x to get
Ÿ
3
x
y'
@
t
D
H
y
@
t
D
+
4
L
2
„
t
=
Ÿ
3
x
t
„
t
,
remembering that
y
@
3
D
= 
9:
Clear
@
x, y, t
D
;
left
=
‡
3
x
y'
@
t
D
H
y
@
t
D
+
4
L
2
„
t
ê
. y
@
3
D
Ø 
9

1
5

1
4
+
y
@
x
D
This is
Mathematica
's way of telling you that
Ÿ
3
x
y'
@
t
D
H
y
@
t
D
+
4
L
2
„
t with
y
@
3
D
= 
9 is:
betterleft
=
1
13

1
4
+
y
@
x
D
1
13

1
4
+
y
@
x
D
Now integrate the right hand side:
right
=
‡
3
x
t
„
t

9
2
+
x
2
2
Here comes the formula for
y
@
x
D
:
Solve
@
betterleft
==
right, y
@
x
DD
::
y
@
x
D
Ø 
2
I

225
+
26 x
2
M

119
+
13 x
2
>>
Compare:
dsol
::
y
@
x
D
Ø 
2
I

81
+
10 x
2
M

43
+
5 x
2
>>
Not bad, eh?
·
B.1.b)
Here is
Mathematica
's formula for the solution of the logistic differential equation
y'
@
x
D
=
a y
@
x
D
J
1

y
@
x
D
b
N
with y
@
0
D
=
5.
Clear
@
y, x, a, b, c, sol
D
;
dsol
=
DSolve
B:
y'
@
x
D
==
a y
@
x
D
1

y
@
x
D
b
,
y
@
0
D
==
5
>
, y
@
x
D
, x
F
::
y
@
x
D
Ø
5 b
‰
a x

5
+
b
+
5
‰
a x
>>
Use the technique of separating and integrating to explain where this formula comes from.
·
Answer:
Rewrite
y'
@
x
D
=
a y
@
x
D
J
1

y
@
x
D
b
N
by separating the variables:
y'
@
x
D
y
@
x
D
I
1

y
@
x
D
b
M
=
a.
Integrate both sides from
0 to
x to get
Ÿ
0
x
y'
@
t
D
y
@
t
D
I
1

y
@
t
D
b
M
„
t
=
Ÿ
0
x
a
„
t,
remembering that
y
@
0
D
=
5:
Clear
@
x, y, t
D
;
left
=
‡
0
x
y'
@
t
D
y
@
t
D I
1

y
@
t
D
b
M
„
t
ê
. y
@
0
D
Ø
5

Log
@
5
D
+
Log
@
5

b
D
+
Log
@
y
@
x
DD

Log
@

b
+
y
@
x
DD
This is
Mathematica
's way of telling you that
Ÿ
0
x
y'
@
t
D
y
@
t
D
I
1

y
@
t
D
b
M
„
t with
y
@
0
D
=
5 is

Log
@
5
D
+
Log
@
5

b
D
+
Log
@
y
@
x
DD

Log
@

b
+
y
@
x
DD
:
Put this in better form, remembering that
Log
@
p
D
+
Log
@
q
D
=
Log
@
p q
D
and

Log
@
r
D
=
Log
A
1
r
E
:
Clear
@
p, q, r, s
D
;
logrules
=
:
Log
@
p
_
D
+
Log
@
q
_
D
Ø
Log
@
p q
D
,

Log
@
r
_
D
Ø
Log
B
1
r
F>
;
betterleft
=
left
êê
. logrules

Log
@
5
D
+
Log
B
H
5

b
L
y
@
x
D

b
+
y
@
x
D
F
Now integrate the right hand side:
right
=
‡
0
x
a
„
t
a x
Now set both sides equal to each other and solve for