Math 1320 Lab3
Instructor: Bridget Fan
Name:
Due: 02/04/16 in class
Show all work including necessary graphs to get full credits!
1. Consider the following initial value problem (IVP):



 dy = y sin(t),
dt


y(0) = −1.
(1)
(a) Verify that y(t) = −e1−cos(t) is a solution of the IVP in Eqn. 1.
(b) Use Euler’s method, with a step size of ∆t = 0.5, to obtain y1 , y2 , y3 and y4 , which
are the estimations of y(x) at t1 = 0.5, t2 = 1.0, t3 = 1.5 and t4 = 2.
Math 1320
Lab3
Page 2 of 7
(c) Calculate the actual value of y(ti ), i = 1, . . . , 4 by plugging the corresponding ti
value into function y(t) from part (a).
yi − y(ti ) , for t = 1, . . . , 4.
(d) Compute the relative error, ei at each point, where ei = y(ti ) (e) Based on the information from part (a) to part (d), filling in the following table.
tn
0
0.5
1
1.5
2
yn
y(tn )
ei
Math 1320
Lab3
Page 3 of 7
2. Match each differential equation with the corresponding vector field and write your answer
down in the table. For each slope field, sketch a few likely solution curves. Use colors!
x−1
dy
=
dx
y
dy
ii.)
= y − sin x
dx
dy
= 1 − y2
dx
dy
iv.)
= x2 + 1
dx
iii.)
i.)
4
4
2
2
0
0
−2
−2
−4
−4
−2
0
2
−4
−4
4
−2
(a) Fig 1
4
2
2
0
0
−2
−2
−4
−4
−4
−4
0
2
4
2
4
(b) Fig 2
4
−2
0
2
4
−2
(c) Fig 3
0
(d) Fig 4
Equation
Fig
i)
ii)
iii)
iv)
Math 1320
Lab3
Page 4 of 7
3. Consider the circuit shown below: a resistor with resistance R, and a capacitor with
capacitance C, are in series with a battery with constant dc voltage V0 .
According to Kirchkoff’s Law, the total voltage drop must equal zero:
−V0 + RI + Q/C = 0,
(2)
where I is the current flowing through the resistor, and Q is the charge on the capacitor.
Charge accumulates on the capacitor due to the current, so
dQ
= I.
dt
Substituting Eqn. 3 into Eqn. 2, and solving for
(3)
dQ
,
dt
dQ
V0
Q
=
−
.
dt
R
RC
yields
(4)
Suppose that R = 10 Ω (ohms), C = 0.1 F (Farads), and V0 = 10 V (volts). Also, suppose
that the gate is open for t < 0, and is then closed at time t = 0.
Math 1320
Lab3
Page 5 of 7
(a) Rewrite the differential equation by plugging in the parameter values.
(b) Sketch the direction field on the grid given below for the equation in part (a)
Q
4
3
2
1
1
2
3
4
5
t
(c) Use a different color to sketch the solution (on top of the direction field) for the
given initial condition, Q(0) = 0.
(d) What is the equilibrium charge on the capacitor? (Hint: The equilibrium occurs
when Q is no longer changing, i.e., when dQ
= 0. Therefore: Let dQ
= 0, and solve
dt
dt
for Q.)
(e) Based on parts (a)–(c), estimate lim Q(t).
t→∞
Math 1320
Lab3
Page 6 of 7
4. A simple model of Earth’s climate can be posed as follows: Incoming energy Ei is balanced
c
by the sum of 1) outgoing energy Eo and 2) the change in energy dE
of the climate.
dt
dEc
dEc
More succinctly: Ei = Eo + dt (or dt = Ei − Eo ).
A central question in the study of the Earth’s climate concerns climate change. In this
c
simple model, climate change is represented by the rate of change of energy: dE
.
dt
Suppose the following:
• The incoming energy Ei is due to incoming solar radiation (minus that which is
reflected). Therefore, Ei = πR2 S0 (1 − α), where S0 = 1362 W m−2 is often called
the “solar constant”, α = 0.3 the albedo, and R is the radius of the Earth.
• All objects radiate energy—in day-to-day life, the energy radiated is not in the
visible spectrum, rather it is infrared light.
Suppose that the radiated energy of the Earth follows the Stefan–Boltzmann law of
the radiation of a blackbody: Eo = 4πR2 σT 4 , where σ = 5.67 × 10−8 W m−2 K−4
is the Stefan–Boltzmann constant.
c
= C dT
, where C = 2.08 × 108 J K−1 m2 is the heat capacity
• Lastly, suppose that dE
dt
dt
of the climate system.
Putting this all together, a differential equation for the average temperature of the Earth
is:
dT
= πR2 S0 (1 − α) − 4πR2 σT 4 ,
(5)
C
dt
with the values of C, S0 , α, and σ given above.
(a) Consider for now a similar (but simplified) version of Eqn. 5:
dy
= 1 − y4.
dx
(6)
Let c be a constant, and verify that the equation 4x − 2 arctan(y(x)) + ln(y(x) − 1) −
ln(y(x) + 1) + c = 0 is an (implicit) solution of Eqn. 6. (Hint: Take the derivative
dy
of the equation, and solve for dx
.)
Math 1320
Lab3
Page 7 of 7
(b) Find the equilibrium temperature of the simple model of Earth’s climate (Eqn. 5).
(Hint: The equilibrium occurs when T is no longer changing, i.e., when dT
= 0.
dt
dT
Therefore, let dt = 0, and solve for T .)
(c) Convert your answer from Kelvin to Fahrenheit.
(d) Does the equilibrium temperature that you found in (b) make sense? If yes, why? If
not, discuss some important components of the climate you think are missing from
this model.