M 340 Assignment #4
Solutions
In each of the following problems write out the governing differential equation, find the
solution to the equation and make the required calculation.
1. If you invest $1000 at an interest rate of 5% per year and make regular monthly deposits
of $150. What will be the value of your account after 30 years?
A t
. 05A t
D A0
Ce .05t 1800
C
. 05
37000e .05t 36000
At
At
37000e 1.5
A 30
36000
1000
1000
D 150 12
1800
37000
. 05
1800
$129, 8 20
2. If you borrow $25000 at an interest rate of 6% per year and make regular payments of
$200 each month, how long will it take to pay off the loan? If you double the monthly
payments, does that cut the length in half?
A t
. 06A t P
A0
25000 P 200 12 2400
At
Ce .06t 2400
. 06
C 25000 2400
25000 40000
15000
. 06
At
15000e .06t 40000
T
40000 15000e .06T 0
40
2. 67
. 06T
15
. 982 /. 06 16. 36yrs
P
400
AT
e .06T
If
so
At
e .06T
T
12
4800 then C
ln 2. 67
25000
0. 982
80000
55000
.06t
80000 55000e
80
1. 45
. 06T
55
. 372 /. 06 6. 2 yrs
ln 1. 45
0. 372
3. Write out the differential equation that models the chemical reaction A B C, letting a
and b denote the initial ammounts of chemicals A and B, respectively. If x t denotes the
ammount of chemical C at time t, sketch the four solution curves that correspond to initial
conditions: x 0
0, 0 x 0
a, a x 0
b, b x 0 .
x t
a
x b
x
1
4. An electric circuit contains a resistance, R, and a capacitance, C, and is driven by an
alternating current E 0 cos t. What is the amplitude of the response, i t , expressed in terms
of R, C, and E 0 ?
Ri t
Ri t
1Qt
C
1it
C
it
i0
it
it
it
cos
Amplitude
E 0 cos t,
E 0 sin t,
i0
0
C E 0 sin t
C 2 R 2 E 0 cos t
2 2 2
C R
1
2
2
E0C R
a
0
C2R2 2 1
E 0 C 2 R 2 e t/RC E C sin t
RC cos t
0
C2R2 2 1
C2R2 2 1
R C cos t
sin t
E0C
E 0 C 2 R 2 e t/RC
C2R2 2 1
C2R2 2 1
C2R2 2 1
E0C
E0C
cos cos t sin sin t
e
2 2 2
2 2 2
R
1
C
C R
1
ae
1
CR
t
R C
C2R2 2
E0C
C2R2 2
sin
1
t/RC
1
C2R2
2
1
1
5.An electric circuit contains a resistance, R, and an inductance, L, and is driven by an
alternating current E 0 cos t. What is the amplitude of the response, i t , expressed in terms
of R, L, and E 0 ?
L di
dt
Ri
it
i0
it
it
cos
Amplitude
E 0 cos t
i0
0
1
RE 0 cos t
L E 0 sin t
Ce L Rt
2 2
2
L
R
RE 0
C 0
L2 2 R2
1
RE 0
L sin t
E 0 R cos t
e L Rt
2
2
2
L
R
L2 2 R2
E0
RE 0
cos cos t sin sin t
e
2
L 2 R2
L2 2 R2
R
L
sin
2 2
2
2 2
L
R
L
R2
E0
2 2
L
R2
1
L
Rt
2