# First Problem Set: Homework Problems

Find the general solutions of the following first-order linear differential equations:

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Find the solutions to the following problems:


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9. Find the general solution of the equation

What happens to all the solutions as ?

Find the solutions of the given differential equations:


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Find the explicit solutions to the initial value problems:


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15. Einsteinium-253 decays at a rate proportional to the amount present. Determine the half-life , if this material loses one third of its mass in 11.7 days.

16. Suppose that 100 mg of thorium-234 are initially present in a closed container, and that thorium-234 is added to the container at a constant rate of 1 mg/day.

(a) Find the amount of thorium-234 in the container at any time, given that its decay rate is 0.02828 days.

(b) Find the limiting amount of thorium-234 in the container as .

(c) How long a time period must elapse before the amount of thorium-234 in the container drops to within 0.5 mg of the limiting value ?

(d) If thorium-234 is added to the container at a rate of mg/day, find the value of that is required tomaintain a constant level of 100 mg of thorium-234.

17. A tank initially contains 120 liters of pure water. A mixture containing g/liter of salt enters the tank at a rate 2 liters/min, and the well-stirred mixture leaves the tank at the same rate. Find an expression in terms of for the amount of salt in the tank at any time . Also, find the limiting amount of salt in the tank as .

18. Suppose that a room containing 120 ft of air is originally free of carbon monoxide. Beginning at time cigarette smoke, containing 4% of carbon monoxide, is introduced to the room at a rate of 0.1 ft/min, and the well-circulated mixture is allowed to leave the room at the same rate.

(a) Find an expression for the concentration of carbon monoxide in the room at any time .

(b) Extended exposure to a carbon monoxide concentration as low as 0.00012 is harmful to the human body. Find the time at which this concentration is reached.

In each of the following two problems sketch versus , determine the equilibrium points and classify each one as stable or unstable:


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21. Semistable Equilibrium Solutions: Sometimes a constant equilibrium solution has the property that solutions lying on one side of the equilibrium solution tend to approach it, whereas solutions lying on the other side recede from it. In this case the equilibrium solution is said to be semistable.

(a) Consider the equation

 (1)

where is a positive constant. Show that is the only equilibrium point, with the corresponding equilibrium solution .

(b) Sketch versus . Show that is increasing as a function of for and also for . Thus solutions below the equilibrium solution approach it while those above it grow further away. Thus is semistable.

(c) Solve Equation (1) subject to the initial condition , and confirm the conclusion reached in part (b).

In each of the following two problems sketch versus and determine the equilibrium points. Also classify each equilibrium point as stable, unstable, or semistable:


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24. Consider the following model of a fishery. Let us assume that the fish are caught at a constant rate independent of the size of the fish population . Then satisfies the equation

 (2)

where and are some positive constants.

(a) If , show that (2) has two equilibrium points and with ; determine these points.

(b) Show that is unstable and is stable.

(c) From a plot of versus show that if the initial population , then as , but if , then decreases as increases. Note that is not an equilibrium point, so if , the extinction will be reached in a finite time.

(d) If , show that decreases to zero as increases regardless of the value of .

(e) If , show that there is a single equilibrium point , and that this equilibrium point is semistable. Notice that is the maximal sustainable yield of the fishery corresponding to the quilibrium value of . The fishery is considered overexploited if is reduced to a level below .

In each of the following problems find the general solution of the given differential equation:


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28. Find the solution of the initial value problem

Sketch the graph of this solution and describe its behavior as increases.

In each of the following problems find the general solution of the given differential equation:


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32. Find the solution of the initial value problem

Sketch the graph of this solution and describe its behavior as increases.

In each of the following two problems find the general solution of the given differential equation:


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Find the solution of the following initial value problems:

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In the following problems, determine the suitable form for the particular solution. You do not need to evaluate the constants.

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In the following three problems, find the general solution to the given differential equation:


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In the following problems, determine the suitable form for the particular solution. You do not need to evaluate the constants.

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In each of the following problems, given one solution, use reduction of order to find the general solution of the given differential equation:

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Find the general solutions of the following Euler's equations


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53. Find the solution of the initial-value problem

Sketch this solution, and discuss its behavior as .

In each of the following problems use variation of parameters to find the general solution of the given differential equation:


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57. Find the general solution of the differential equation

given that two linearly independent solutions of the homogeneous equations are

In each of the following two problems determine , , and so as to write the given expression in the form


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60. A mass weighing 16 lb stretches a spring 3 in. The mass is attached to a viscous damper with a damping constant of 2 lb-sec/ft. If the mass is set in motion from its equilibrium position with a downward velocity of 3 in/sec, find its position at any time . Draw versus . Determine when the mass first returns to its equilibrium position. Also, find the time such that in for all . (Recall that ft/sec.)

The positions of certain mass-spring systems satisfy the following initial value problems


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(a) Find the solutions of these initial value problems.

(b) Draw versus and versus on the same axes. (c) Draw versus ; that is, draw and parametrically with as the parameter. What is the direction on this curve as increases? Identify several corresponding points on the curves in parts (b) and (c).

63. A mass of 5 kg stretches a spring 10 cm. The mass is acted on by an external force of N (newtons=kgm/sec) and moves in a medium that imparts a viscous force of 2 N when the speed is of the mass is 4 cm/sec.

(a) If the mass is set in motion from its equilibrium position with an initial velocity of 3 cm/sec, formulate the initial value problem describing the motion of the mass, and find its solution.

(b) Identify the transient and steady-state parts of the solution.

(c) Draw the graph of the solution, as well as the steady-state solution.

(d) If the given external source is replaced by a force of frequency , find the value of for which the amplitude of the forced response is maximum.

64. Consider a vibrating system described by the initial value problem

(a) Determine the steady part of the solution of this problem.

(b) Find the amplitude of the steady state solution in terms of .

(c) Find the maximum value of and the frequency for which it occurs.

(d) Draw versus .

In each of the following problems, find the eigenvalues and eigenfunctions of the given boundary-value problem. Assume that all eigenvalues are real.


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In each of the following problems, find the Fourier series corresponding to the given function:

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72. If for and , find a formula for in the interval , and in the interval .

Find the Fourier series for the following problems. Assume that the functions are periodically extended outside the original interval. Sketch the function to which each series converges outside the original interval.

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In each of the following problems find the required Fourier series for the given function; sketch the graph of the function to which the function converges over two or three periods.

76. Both even and odd extensions, period 4.

77. Sine series, period 4.

78. Cosine series, period .

79. Sine series, period .

80. Cosine series, period .

81. Sine series, period .

82. State exactly the boundary value problem determining the temperature in a silver rod 2 meters long if the ends are held at the temperatures 30 Celsius and 50 Celsius, respectively. The thermal diffusivity of silver is cmsec. Assume that the initial temperature in the bar is given by a quadratic function of the distance along the bar consistent with the preceding boundary conditions, and with the condition that the temperature at the center of the rod is 60 Celsius.

83. Find the solution of the heat conduction problem

In each of the following problems determine whether the method of separation of variables can be used to replace the given partial differential equation by a pair of ordinary differential equations. If so, find the equations.


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87. Consider the conduction of heat in a copper rod 100 cm in length whose ends are maintained at 0 Celsius for all . Find an expression for the temperature if the initial temperature distribution in the rod is given by

88. Let a metallic rod 20 cm long be heated to a uniform temperature of 100 Celsius. Suppose that at the ends of the bar are plunged into an ice bath at 0 Celsius, and thereafter maintained at this temperature, but that no heat is allowed to escape through the lateral surface. Find an expression for the temperature at any point in the bar at any later time. Use two terms in the series expansion to determine approximately the temperature at the center of the bar at time sec if the bar is made of (a) silver, (b) aluminum, or (c) cast iron. Thermal diffusivities of silver, aluminum, and cast iron are 1.71 cm/sec, 0.86 cm/sec, and 0.12 cm/sec, respectively. Also, use just one term in the series expansion for to find the time that will elapse before the center of the bar cools to a temperature of 25 Celsius for each of the three metals.

89. Let an aluminum rod of length be initially at the uniform temperature of 25 Celsius. Suppose that at time the end is cooled to 0 Celsius, while the end is heated to 60 Celsius, and both are thereafter maintained at those temperatures.
(a) Find the temperature distribution in the rod at any time .
Assume in the remaining parts of this problem that cm.
(b) Use only the first term in the series for the temperature to find the approximate temeprature at cm when sec; when sec.
(c) Use the first two terms in the series for to find an approximate value of . What is the percentage difference between the one- and -two-term approximations? Does the third term have any appreciable effect for this value of ?
(d) Use the first term in the series for to estimate the time interval that must elapse before the temperature at cm comes within 1 percent of its steady state value.

90. Consider a uniform rod of length with an initial temperature given by , . Assume that both ends of the bar are insulated. Find the temperature , and the steady-state temperature as .

In each of the following problems find the steady-state solution of the heat conduction equation that satisfies the given set of boundary conditions.


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94. Find the steady-state solution of the partial differential equation

that satisfies the boundary conditions

95. Consider a uniform bar of length having an initial temperature distribution given by , . Assume that the temperature at the end is held at 0 Celsius, while the end is insulated so that no heat passes through it.
(a) Show that the fundamental solutions of the partial differential equation and boundary conditions are

(b) Find a formal series expansion for the temperature ,

that also satisfies the initial condition .

96. Find the displacement in an elastic string, fixed at both ends, that is set in motion with no initial velocity from the initial position , where

97. Find the displacement in an elastic string of length , fixed at both ends, that is set in motion from its straight equilibrium position with the initial velocity defined by

98. If an elastic string is free at one end, the boundary condition there is that . Find the displacement in an elastic string of length , fixed at and free at , set in motion with no inital velocity from the initial position , where is a given function.
HINT: Show that the fundamental solutions for this problem, satisfying all conditions except the inhomogeneous initial condition, are

99. A vibrating string moving in an elastic medium satisfies the equation

where is proportional to the coefficient of elasticity of the medium. Suppose that the string is fixed at the ends, and is released with no initial velocity from the initial position , . Find the displacement .

100. The motion of a circular elastic membrane, such as a drum head, is governed by the two-dimensional wave equation in polar coordinates

Assuming that , find ordinary differential equations satisfied by , and . (Do not solve them.)

In each of the following problems find the general solution of the given system of equations. Also, sketch a number of representative trajectories.


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In each of the following problems find all the equilibrium points, determine their type and stability, and sketch local phase portraits near them. If possible, also try to sketch the global phase portraits in the whole plane.

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Dr Yuri V Lvov 2016-04-22