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2. (a) In the theory of learning, the rate at which a subject is memorized is assumed to be proportional to the amount that is left to be memorized. Suppose M denotes the total amount of a subject to be memorized and A(t) is the amount memorized in time t. Determine a differential equation for the amount A(t). (SET UP ONLY. DO NOT SOLVE.) (b) in time t. Determine a differential equation for the amount A(t) when forgetfulness is taken into account. (SET UP ONLY. DO NOT SOLVE.) Now assume that the rate at which material is forgotten is proportional to the amount memorized

2. (a) In the theory of learning, the rate at which a subject is memorized is assumed to be proportional to the amount that is left to be memorized. Suppose M denotes the total amount of a subject to be memorized and A(t) is the amount memorized in time t. Determine a differential equation for the amount A(t). (SET UP ONLY. DO NOT SOLVE.) (b) in time t. Determine a differential equation for the amount A(t) when forgetfulness is taken into account. (SET UP ONLY. DO NOT SOLVE.) Now assume that the rate at which material is forgotten is proportional to the amount memorized

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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1. Use GeoGebra to create a direction field for the differential equation \(\frac{dy}{dx} = x^2 - y\). Then use GeoGebra to draw an approximate solution curve that passes through the indicated points. (This will have 4 different curves on one picture. No written work needed here, just submit your screenshot/graphics file.) - (a) \( y(-6) = 0 \) - (b) \( y(0) = 1 \) - (c) \( y(0) = -4 \) - (d) \( y(8) = -4 \) 2. - (a) In the theory of learning, the rate at which a subject is memorized is assumed to be proportional to the amount that is left to be memorized. Suppose \( M \) denotes the total amount of a subject to be memorized and \( A(t) \) is the amount memorized in time \( t \). Determine a differential equation for the amount \( A(t) \). (SET UP ONLY. DO NOT SOLVE.) - (b) Now assume that the rate at which material is forgotten is proportional to the amount memorized in time \( t \). Determine a differential equation for the amount \( A(t) \) when forgetfulness is taken into account. (SET UP ONLY. DO NOT SOLVE.) 3. Find the critical points and phase portrait of the autonomous first-order differential equation \(\frac{dy}{dx} = y^2(y+4)(y-2)\). Classify each critical point as an attractor (asymptotically stable), repeller (unstable), or semi-stable. 4. Solve the IVP \(\frac{dy}{dt} + 2(t+1)y^2 = 0\), \( y(0) = -\frac{1}{8} \). 5. The temperature of a cup of coffee obeys Newton’s law of cooling. The initial temperature of the coffee is \( 200^\circ F \) and one minute later, it is \( 180^\circ F \). The ambient temperature of the room is \( 66^\circ F \). - (a) If \( T(t) \) represents the temperature of the coffee at time \( t \), write the initial value problem that represents this scenario.
Transcribed Image Text:1. Use GeoGebra to create a direction field for the differential equation \(\frac{dy}{dx} = x^2 - y\). Then use GeoGebra to draw an approximate solution curve that passes through the indicated points. (This will have 4 different curves on one picture. No written work needed here, just submit your screenshot/graphics file.) - (a) \( y(-6) = 0 \) - (b) \( y(0) = 1 \) - (c) \( y(0) = -4 \) - (d) \( y(8) = -4 \) 2. - (a) In the theory of learning, the rate at which a subject is memorized is assumed to be proportional to the amount that is left to be memorized. Suppose \( M \) denotes the total amount of a subject to be memorized and \( A(t) \) is the amount memorized in time \( t \). Determine a differential equation for the amount \( A(t) \). (SET UP ONLY. DO NOT SOLVE.) - (b) Now assume that the rate at which material is forgotten is proportional to the amount memorized in time \( t \). Determine a differential equation for the amount \( A(t) \) when forgetfulness is taken into account. (SET UP ONLY. DO NOT SOLVE.) 3. Find the critical points and phase portrait of the autonomous first-order differential equation \(\frac{dy}{dx} = y^2(y+4)(y-2)\). Classify each critical point as an attractor (asymptotically stable), repeller (unstable), or semi-stable. 4. Solve the IVP \(\frac{dy}{dt} + 2(t+1)y^2 = 0\), \( y(0) = -\frac{1}{8} \). 5. The temperature of a cup of coffee obeys Newton’s law of cooling. The initial temperature of the coffee is \( 200^\circ F \) and one minute later, it is \( 180^\circ F \). The ambient temperature of the room is \( 66^\circ F \). - (a) If \( T(t) \) represents the temperature of the coffee at time \( t \), write the initial value problem that represents this scenario.
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