Chapter 9

Differential Equations

Educators

EN
Jm
+ 3 more educators

Problem 1

Show that $ y = \frac{2}{3}e^x + e^{-2x} $ is a solution of the differential equation $ y^x + 2y = 2e^x $.

Mikyla S.
Numerade Educator

Problem 2

Verify that $ y = -t \cos t - t $ is a solution of the initial value problem.

$ t \frac{ty}{dt} = y + t^{2} \sin t $ $ y = \pi = 0 $

Mikyla S.
Numerade Educator

Problem 3

(a) For what values of $ r $ does the function $ y = e^{rx} $ satisfy the differential equation $ 2y^{"} + y^{'} - y = 0? $

(b) if $ r_1 $ and $ r_2 $ are the values of $ r $ that you found in part (a), show that every member of the family of functions $ y = ae^{r_1{x}} + be^{r_2{x}} $ is also a solution.

Mikyla S.
Numerade Educator

Problem 4

(a) For what values of $ k $ does the function $ y = \cos kt $ satisfy the differential equation $ 4y^{"} = - 25y? $

(b) For those values of $ k. $ verify that every member of the family functions $ y = A \sin kt + B \cos kt $ is also a solution.

Mikyla S.
Numerade Educator

Problem 5

Which of the following functions are solutions of the differential equation $ y^{"} + y = \sin x ? $

(a) $ y = \sin x $
(b) $ y = \cos x $
(c) $ y = \frac {1}{2} x \sin x $
(d) $ y = - \frac{1}{2} x \cos x $

Mikyla S.
Numerade Educator

Problem 6

(a) Show that every member of the family of functions $ y = $ (In $ x + C)/x $ is a solution of the differential equation $ x^{2} y^{'} + xy = 1. $

(b) Illustrate part (a) by graphing several members of the family of solutions on a common screen.

(c) Find a solution of the differential equation that satisfies the initial condition. $ y(1) = 2. $

(d) Find a solution of the differential equation that satisfies the initial condition . $ y(2) = 2. $

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Problem 7

(a) What can you say about a solution of the equation $ y{'} = - y^{2} $ just by looking at the differential equation?

(b) Verify that all members of the family $ y = 1/(x + C) $ are solutions of the equation in part (a).

(c) Can you think of a solution of the differential equation $ y^{'} = - y^{2} $ that is not a member of the family in part (b)?

(d) Find a solution of the initial-value problem.

$ y^{'} = - y^2 $ $ y (0) = 0.5 $

Jm
John M.
Numerade Educator

Problem 8

(a) What can you say about the graph of a solution of the equation $ y^{'} = xy^3 $ when $ x $ is close to $ 0? $ What if $ x $ is large?

(b) Verify that all members of the family $ y = (c - x^2) ^{{-}{1/2}} $ are solutions of the differential equation $ y^{'} = xy^3. $

(c) Graph several members of the family of solutions on a common screen. Do the graphs confirm what you predicted in part (a)?

(d) Find a solution of the initial-value problem

$ y^{'} = xy^3 $ $ y(0) = 2 $

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Problem 9

A population is modeled by the differential equation

$ \frac {dP}{dt} = 1.2 P \left(1 - \frac{P}{4200} \right) $

(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?

EN
Elena N.
University of Utah

Problem 10

The Fitzhugh-Nagumo model for the electrical impulse in a neuron states that, in the absence of relaxation effects, the electrical potential in a neuron $ v(t) $ obeys the differential equation

$ \frac {dv}{dt} = - v [v^2 - (1 + a) v + a] $

where $ a $ is a positive constant such that $ 0 < a < 1. $
(a) For what values of $ v $ is $ v $ unchanging (that is, $ dv/dt = 0)?. $
(b) For what values of $ v $ is $ v $ increasing?
(c) For what values of $ v $ is $ v $ decreasing?

Paul A.
California State Polytechnic University, Pomona

Problem 11

Explain why the functions with the given graphs can't be solutions of the differential equations

$ \frac {dy}{dt} = e^t (y - 1)^2 $

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Problem 12

The function with the given graph is a solution of one of the following differential equations. Decide which is the correct equation and justify your answer.

A. $ y^{'} = 1 + xy $
B. $ y^{'} = -2 xy $
C. $ y^{'} = 1 - 2xy $

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Problem 13

Match the differential equations with the solution graphs labeled I-IV. Give reasons for your choices.

(a) $ y^{'} = 1 + x^{2} + y^{2} $

(b) $ y^{'} = xe^{-x^2 -y^2} $

(c) $ y^{'} = \frac {1} {1 + e^{{x^2} + {y^2}}} $

(d) $ y^{'} = \sin (xy) \cos (xy) $

Alexander W.
University of Missouri - Columbia

Problem 14

Suppose you have just poured a cup of freshly brewed coffee with temperature $ 95^{\circ} $ in a room where the temperature is $ 20^{\circ}. $

(a) When do you think the coffee cools most quickly? What happens to the rate of cooling as time goes by? Explain.

(b) Newtons Law of Cooling states that the rate of cooling of an object is proportional to the temperature difference between the object and its surroundings, provided that this difference is not too large. Write a differential equation that expresses Newtons Law of Cooling for this particular situation. What is the initial condition? In view of your answer to part (a), do you think this differential equation is an appropriate model for cooling?

(c) Make a rough sketch of the graph of the solution of the initial-value problem in part (b).

KA
Kaci A.
Numerade Educator

Problem 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 $ increase? Explain.

(b) If $ M $ is the maximum level of performance of which the learner is capable, explain why the differential equation

$ \frac {dP}{dt} = k(M - P) $ $ K $ a positive constant

Bobby B.
University of North Texas

Problem 16

Von Bertalanffys equation states that the rate of growth in length of an individual fish is proportional to the difference between the current lenght $ L $ and the asymptotic length $ L_x $ (in centimeters).
(a) Write a differential equation that expresses this idea.
(b) Make a rough sketch of the graph of a solution of a typical initial-value problem for this differential equation.

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Problem 17

Differential equations have been used extensively in the study of drug dissolution for patients given oral medication. One such equation is the Weibull equation for the concentration c(t) of the drug:

$ \frac {dc}{dt} = \frac {k}{t^b} (c_a - c) $
where $ k $ and $ c_a $ are positive constants and $ 0 < b < 1. $ Verify that

$ c(t) = c_a (1 - e^{-at^{1-b}}) $
is a solution of the Weibull equation for $ 1 > 0, $ where $ a = k/(1 - b). $ What does the differential equation say about how drug dissolution occurs?

Vishal P.
Numerade Educator