Exam (elaborations)
Problem Set #3.
Problem Set #3. Problem Set #3. Problem Set #3.
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Problem Set #3Sections 9.1 – 9.6
Due Monday, October 28th, 2019
1. (9.1.2) Verify that 𝑦 = −𝑡 cos(𝑡) − 𝑡 is a solution of the initial-value problem
𝑑𝑦
𝑡 = 𝑦 + 𝑡 2 sin(𝑡) , 𝑦(𝜋) = 0
𝑑𝑡
2. (9.1.9) A population is modeled by the differential equation
𝑑𝑃 𝑃
= 1.2𝑃 (1 − )
𝑑𝑡 4200
(a) For what values of 𝑃 is the population increasing?
(b) For what values of 𝑃 is the population decreasing?
(c) What are the equilibrium solutions?
3. (9.1.15) Psychologists interested in learning theory study learning curves. A learning curve is the
graph of a function 𝑃(𝑡), the performance of someone learning a skill as a function of the
training time 𝑡. The derivative 𝑑𝑃/𝑑𝑡 represents the rate at which performance improves.
(a) When do you think 𝑃 increases most rapidly? What happens to 𝑑𝑃/𝑑𝑡 as 𝑡 increases?
Explain.
(b) If 𝑀 is the maximum level of performance of which the learner is capable, explain why the
differential equation
𝑑𝑃
= 𝑘(𝑀 − 𝑃)
𝑑𝑡
where 𝑘 is a positive constant, is a reasonable model for learning.
(c) Make a rough sketch of a possible solution of this differential equation.
4. (9.2.11) Sketch the direction field of the differential equation. Then use it to sketch a solution
curve that passes through the given point.
𝑦 ′ = 𝑦 − 2𝑥, (1, 0)
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, 5. (9.2.22) Use Euler’s method with step size 0.2 to estimate 𝑦(1), where 𝑦(𝑥) is the solution of
the initial-value problem
1
𝑦′ = 𝑥2𝑦 − 𝑦2, 𝑦(0) = 1
2
6. (9.2.27) The figure shows a circuit containing an electromotive force, a capacitor with
capacitance of 𝐶 farads (F), and a resistor with resistance of 𝑅 ohms (Ω). The voltage drop across
the capacitor is 𝑄/𝐶, where 𝑄 is the charge (in Coulombs, C), so in this case Kirchohff’s Law
gives
𝑄
𝑅𝐼 + = 𝐸(𝑡)
𝐶
𝑑𝑄
But 𝐼 = 𝑑𝑡
, so we have
𝑑𝑄 1
𝑅 + 𝑄 = 𝐸(𝑡)
𝑑𝑡 𝐶
Suppose the resistance is 5 Ω, the capacitance is 0.05 F, and a battery gives a constant voltage of 60 V.
(a) Draw a direction field for this differential equation.
(b) What is the limiting value of the charge?
(c) Is there an equilibrium solution?
(d) If the initial charge is 𝑄(0) = 0 C, use the direction field to sketch the solution curve.
(e) If the initial charge is 𝑄(0) = 0 C, use Euler’s method with step size 0.1 to estimate the
charge after half a second.
7. (9.3.7) Solve the differential equation.
𝑑𝜃 𝑡 sec(𝜃)
= 2
𝑑𝑡 𝜃𝑒 𝑡
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