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Math 1320-6 Lab 3 Name: T.A.: Kyle Steffen 28 January 2016 uNID: Instructions and due date: • Due: 4 February 2016 at the start of class. • For full credit: Show all of your work (including any sketches), and simplify your final answers. • Work together! However, your work should be your own (not copied from a group member). 1. Consider the following initial value problem (IVP): dy = y sin(t), dt y(0) = −1. (1) (a) Verify that y = −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 an estimate of y(2) (where y(2) is the solution of the IVP given by Eqn. 1). Math 1320-6 Lab 3 - Page 2 of 7 (c) Denote t0 = 0, t1 = 0.5, t2 = 1.0, t3 = 1.5, and t4 = 2.0. Then y0 = −1, y1 is the approximation (computed by Euler’s method) of y(t1 ), y2 is the approximation (computed by Euler’s method) of y(t2 ), etc. Compute the relative error, ei , of each yi − y(ti ) , for t = 0, . . . , 4. approximation found in part (b): ei = y(ti ) (d) Make a table with four columns (tn , yn , y(tn ), ei ), and collect the data from parts (b)–(c). Math 1320-6 Lab 3 - Page 3 of 7 2. Match each differential equation with the corresponding vector field. For each slope field, sketch a few likely solution curves. Use color! 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 (c) Fig 3 2 4 2 4 (b) Fig 2 4 −2 0 2 4 −2 0 (d) Fig 4 Math 1320-6 Lab 3 - 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 1320-6 Lab 3 - Page 5 of 7 (a) Sketch the direction field for Eqn. 4 with the given parameters. (b) Sketch the solution (on top of the direction field) with the given initial condition. (Hint: Q(0) = 0.) Q 3 2 1 1 2 3 4 5 t (c) 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.) (d) Based on parts (a)–(c), estimate lim Q(t). t→∞ Math 1320-6 Lab 3 - 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 1320-6 Lab 3 - 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 .) Convert your answer from Kelvin to Fahrenheit. (c) 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.