Differential Equations
In physics, engineering, chemistry, economics, and other sciences mathematical models
are built that involve rates at which things happen. These models are equations and the
rates are derivatives. Equations containing derivatives are called
differential equations
.
Examples
:
1) 100 grams of cane sugar in water are being converted into dextrose at a rate that is
proportional to the amount unconverted. Find the differential equation expressing the rate
of conversion after t minutes.
Solution
:
Let q be the grams of sugar converted in t minutes, then (100 – q) is the number of grams
unconverted and the rate of conversion is given by
dq
dt
=
k(100
!
q)
where k is the
constant of proportionality.
2) A curve is defined by the condition that at each of its points (x,y), is slope dy/dx is
equal twice the sum of the coordinates of the point, find the differential equation that
defines the curve.
Solution
:
dy
dx
=
2(x
+
y)
The following are more examples of differential equations:
1)
dy
dx
=
cosx
or
y'
=
cosx
2)
d
2
y
dx
2
+
k
3
y
=
0
or
!
!
y
+
k
3
y
=
0
3)
(x
2
+
y
2
)dx
=
2xdy
4)
!
u
!
t
=
h
2
(
!
2
u
!
x
2
+
!
2
u
!
y
2
)
5)
!
2
w
!
x
2
+
!
w
!
y
=
0
6)
d
2
s
dt
2
!
"
#
$
%
&
6
’
xy
ds
dt
+
s
=
0
7)
d
3
x
dy
3
+
x
dx
dy
=
4xy
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8)
d
5
y
dx
5
+
x
dy
dx
!
"
#
$
%
&
3
’
8y
=
0
9)
d
2
y
dt
2
+
x
d
2
x
dt
2
=
x
10)
x
!
f
!
x
+
y
!
f
!
y
=
nf
11)
L
d
2
u
dt
2
+
R
du
dt
+
1
c
u
=
Ecost
12)
c
!
!
!
y
+
m
!
!
y
"
y
=
6
Definition
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