In Problems 1 and 2 fill in the blank and then write this result as a linear first-order differential equation that is free of the symbol c1 and has the form dydx f(x, y). The symbol c1 represents a constant.
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Table of Contents
1
INTRODUCTION TO DIFFERENTIAL EQUATIONS
1.1
Definitions and Terminology
1.2
Initial-Value Problems
1.3
Differential Equations as Mathematical Models
2
FIRST-ORDER DIFFERENTIAL EQUATIONS
2.1
Solution Curves Without a Solution
2.2
Separable Variables
2.3
Linear Equations
2.4
Exact Equations
2.5
Solutions by Substitutions
2.6
A Numerical Method
3
MODELING WITH FIRST-ORDER DIFFERENTIAL EQUATIONS
3.1
Linear Models
3.2
Nonlinear Models
3.3
Modeling with Systems of First-Order DEs
4
HIGHER-ORDER DIFFERENTIAL EQUATIONS
4.1
Preliminary TheoryLinear Equations
4.2
Reduction of Order
4.3
Homogeneous Linear Equations with Constant Coefficients
4.4
Undetermined CoefficientsSuperposition Approach
4.5
Undetermined CoefficientsAnnihilator Approach
4.6
Variation of Parameters
4.7
Cauchy-Euler Equation
4.8
Solving Systems of Linear DEs by Elimination
4.9
Nonlinear Differential Equations
5
MODELING WITH HIGHER-ORDER DIFFERENTIAL EQUATIONS
5.1
Linear Models: Initial-Value Problems
5.2
Linear Models: Boundary-Value Problems
5.3
Nonlinear Models
6
SERIES SOLUTIONS OF LINEAR EQUATIONS
6.1
Solutions About Ordinary Points
6.2
Solutions About Singular Points
6.3
Special Functions
7
THE LAPLACE TRANSFORM
7.1
Definition of the Laplace Transform
7.2
Inverse Transforms and Transforms of Derivatives
7.3
Operational Properties I
7.4
Operational Properties II
7.5
The Dirac Delta Function
7.6
Systems of Linear Differential Equations
8
SYSTEMS OF LINEAR FIRST-ORDER DIFFERENTIAL EQUATIONS
8.1
Preliminary TheoryLinear Systems
8.2
Homogeneous Linear Systems
8.3
Nonhomogeneous Linear Systems
8.4
Matrix Exponential
9
NUMERICAL SOLUTIONS OF ORDINARY DIFFERENTIAL EQUATIONS
9.1
Euler Methods and Error Analysis
9.2
Runge-Kutta Methods
9.3
Multistep Methods
9.4
Higher-Order Equations and Systems
9.5
Second-Order Boundary-Value Problems
10
PLANE AUTONOMOUS SYSTEMS
10.1
Autonomous Systems
10.2
Stability of Linear Systems
10.3
Linearization and Local Stability
10.4
Autonomous Systems as Mathematical Models
11
ORTHOGONAL FUNCTIONS AND FOURIER SERIES
11.1
Orthogonal Functions
11.2
Fourier Series
11.3
Fourier Cosine and Sine Series
11.4
Sturm-Liouville Problem
11.5
Bessel and Legendre Series
12
BOUNDARY-VALUE PROBLEMS IN RECTANGULAR COORDINATES
12.1
Separable Partial Differential Equations
12.2
Classical PDEs and Boundary-Value Problems
12.3
Heat Equation
12.4
Wave Equation
12.5
Laplaces Equation
12.6
Nonhomogeneous Boundary-Value Problems
12.7
Orthogonal Series Expansions
12.8
Higher-Dimensional Problems
13
BOUNDARY-VALUE PROBLEMS IN OTHER COORDINATE SYSTEMS
13.1
Polar Coordinates
13.2
Polar and Cylindrical Coordinates
13.3
Spherical Coordinates
14
INTEGRAL TRANSFORMS
14.1
Error Function
14.2
Laplace Transform
14.3
Fourier Integral
14.4
Fourier Transforms
15
NUMERICAL SOLUTIONS OF PARTIAL DIFFERENTIAL EQUATIONS
15.1
Laplaces Equation
15.2
Heat Equation
15.3
Wave Equation
Textbook Solutions for Differential Equations with Boundary-Value Problems
Chapter 1 Problem 32
Question
A tank in the form of a right-circular cylinder of radius 2 feet and height 10 feet is standing on end. If the tank is initially full of water and water leaks from a circular hole of radius inch at its bottom, determine a differential equation for the height h of the water at time t. Ignore friction and contraction of water at the hole.
Solution
The first step in solving 1 problem number 32 trying to solve the problem we have to refer to the textbook question: A tank in the form of a right-circular cylinder of radius 2 feet and height 10 feet is standing on end. If the tank is initially full of water and water leaks from a circular hole of radius inch at its bottom, determine a differential equation for the height h of the water at time t. Ignore friction and contraction of water at the hole.
From the textbook chapter INTRODUCTION TO DIFFERENTIAL EQUATIONS you will find a few key concepts needed to solve this.
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Title
Differential Equations with Boundary-Value Problems 7
Author
Dennis G. Zill, Michael R. Cullen
ISBN
9780495108368
Solution: A tank in the form of a right-circular cylinder of radius 2 feet and height 10
Chapter 1 textbook questions
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Chapter 1: Problem 1 Differential Equations with Boundary-Value Problems 7
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Chapter 1: Problem 2 Differential Equations with Boundary-Value Problems 7
In Problems 1 and 2 fill in the blank and then write this result as a linear first-order differential equation that is free of the symbol c1 and has the form dydx f(x, y). The symbol c1 represents a constant.
Read more -
Chapter 1: Problem 3 Differential Equations with Boundary-Value Problems 7
In Problems 3 and 4 fill in the blank and then write this result as a linear second-order differential equation that is free of the symbols c1 and c2 and has the form F(y, y ) 0. The symbols c1, c2, and k represent constants.
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Chapter 1: Problem 4 Differential Equations with Boundary-Value Problems 7
In Problems 3 and 4 fill in the blank and then write this result as a linear second-order differential equation that is free of the symbols c1 and c2 and has the form F(y, y ) 0. The symbols c1, c2, and k represent constants.
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Chapter 1: Problem 5 Differential Equations with Boundary-Value Problems 7
In Problems 5 and 6 compute y and y and then combine these derivatives with y as a linear second-order differential equation that is free of the symbols c1 and c2 and has the form F(y, y y ) 0. The symbols c1 and c2 represent constants.
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Chapter 1: Problem 6 Differential Equations with Boundary-Value Problems 7
In Problems 5 and 6 compute y and y and then combine these derivatives with y as a linear second-order differential equation that is free of the symbols c1 and c2 and has the form F(y, y y ) 0. The symbols c1 and c2 represent constants.
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Chapter 1: Problem 7 Differential Equations with Boundary-Value Problems 7
In Problems 712 match each of the given differential equations with one or more of these solutions: (a) y 0, (b) y 2, (c) y 2x, (d) y 2x2 .
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Chapter 1: Problem 8 Differential Equations with Boundary-Value Problems 7
In Problems 712 match each of the given differential equations with one or more of these solutions: (a) y 0, (b) y 2, (c) y 2x, (d) y 2x2 .
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Chapter 1: Problem 9 Differential Equations with Boundary-Value Problems 7
In Problems 712 match each of the given differential equations with one or more of these solutions: (a) y 0, (b) y 2, (c) y 2x, (d) y 2x2 .
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Chapter 1: Problem 10 Differential Equations with Boundary-Value Problems 7
In Problems 712 match each of the given differential equations with one or more of these solutions: (a) y 0, (b) y 2, (c) y 2x, (d) y 2x2 .
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Chapter 1: Problem 11 Differential Equations with Boundary-Value Problems 7
In Problems 712 match each of the given differential equations with one or more of these solutions: (a) y 0, (b) y 2, (c) y 2x, (d) y 2x2 .
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Chapter 1: Problem 12 Differential Equations with Boundary-Value Problems 7
In Problems 712 match each of the given differential equations with one or more of these solutions: (a) y 0, (b) y 2, (c) y 2x, (d) y 2x2 .
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Chapter 1: Problem 13 Differential Equations with Boundary-Value Problems 7
In Problems 13 and 14 determine by inspection at least one solution of the given differential equation.
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Chapter 1: Problem 14 Differential Equations with Boundary-Value Problems 7
In Problems 13 and 14 determine by inspection at least one solution of the given differential equation.
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Chapter 1: Problem 15 Differential Equations with Boundary-Value Problems 7
In Problems 15 and 16 interpret each statement as a differential equation.
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Chapter 1: Problem 16 Differential Equations with Boundary-Value Problems 7
In Problems 15 and 16 interpret each statement as a differential equation.
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Chapter 1: Problem 17 Differential Equations with Boundary-Value Problems 7
(a) Give the domain of the function y x2/3. (b) Give the largest interval I of definition over which y x2/3 is solution of the differential equation 3xy 2y 0.
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Chapter 1: Problem 18 Differential Equations with Boundary-Value Problems 7
(a) Verify that the one-parameter family y2 2y x2 x c is an implicit solution of the differential equation (2y 2)y 2x 1. (b) Find a member of the one-parameter family in part (a) that satisfies the initial condition y(0) 1. (c) Use your result in part (b) to find an explicit function y (x) that satisfies y(0) 1. Give the domain of the function . Is y (x) a solution of the initial-value problem? If so, give its interval I of definition; if not, explain.
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Chapter 1: Problem 19 Differential Equations with Boundary-Value Problems 7
Given that y x 2x is a solution of the DE xy y 2x. Find x0 and the largest interval I for which y(x) is a solution of the first-order IVP xy y 2x, y(x0) 1.
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Chapter 1: Problem 20 Differential Equations with Boundary-Value Problems 7
Suppose that y(x) denotes a solution of the first-order IVP y x2 y2 , y(1) 1 and that y(x) possesses at least a second derivative at x 1. In some neighborhood of x 1 use the DE to determine whether y(x) is increasing or decreasing and whether the graph y(x) is concave up or concave down.
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Chapter 1: Problem 21 Differential Equations with Boundary-Value Problems 7
A differential equation may possess more than one family of solutions. (a) Plot different members of the families y 1(x) x2 c1 and y 2(x) x2 c2. (b) Verify that y 1(x) and y 2(x) are two solutions of the nonlinear first-order differential equation (y)2 4x2 . (c) Construct a piecewise-defined function that is a solution of the nonlinear DE in part (b) but is not a member of either family of solutions in part (a).
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Chapter 1: Problem 22 Differential Equations with Boundary-Value Problems 7
What is the slope of the tangent line to the graph of a solution of that passes through (1, 4)? In Pr
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Chapter 1: Problem 23 Differential Equations with Boundary-Value Problems 7
In Problems 2326 verify that the indicated function is a particular solution of the given differential equation. Give an interval of definition I for each solution.
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Chapter 1: Problem 24 Differential Equations with Boundary-Value Problems 7
In Problems 2326 verify that the indicated function is a particular solution of the given differential equation. Give an interval of definition I for each solution.
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Chapter 1: Problem 25 Differential Equations with Boundary-Value Problems 7
In Problems 2326 verify that the indicated function is a particular solution of the given differential equation. Give an interval of definition I for each solution.
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Chapter 1: Problem 26 Differential Equations with Boundary-Value Problems 7
In Problems 2326 verify that the indicated function is a particular solution of the given differential equation. Give an interval of definition I for each solution.
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Chapter 1: Problem 27 Differential Equations with Boundary-Value Problems 7
In Problems 2730, y c1e3x c2ex 2x is a twoparameter family of the second-order DE y 2y 3y 6x 4. Find a solution of the second-order IVP consisting of this differential equation and the given initial conditions. 2
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Chapter 1: Problem 28 Differential Equations with Boundary-Value Problems 7
In Problems 2730, y c1e3x c2ex 2x is a twoparameter family of the second-order DE y 2y 3y 6x 4. Find a solution of the second-order IVP consisting of this differential equation and the given initial conditions. 2
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Chapter 1: Problem 29 Differential Equations with Boundary-Value Problems 7
In Problems 2730, y c1e3x c2ex 2x is a twoparameter family of the second-order DE y 2y 3y 6x 4. Find a solution of the second-order IVP consisting of this differential equation and the given initial conditions. 2
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Chapter 1: Problem 30 Differential Equations with Boundary-Value Problems 7
In Problems 2730, y c1e3x c2ex 2x is a twoparameter family of the second-order DE y 2y 3y 6x 4. Find a solution of the second-order IVP consisting of this differential equation and the given initial conditions. 2
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Chapter 1: Problem 31 Differential Equations with Boundary-Value Problems 7
The graph of a solution of a second-order initial-value problem d2ydx2 f(x, y, y), y(2) y0, y(2) y1, is given in Figure 1.R.1. Use the graph to estimate the values of y0 and y1.
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Chapter 1: Problem 32 Differential Equations with Boundary-Value Problems 7
A tank in the form of a right-circular cylinder of radius 2 feet and height 10 feet is standing on end. If the tank is initially full of water and water leaks from a circular hole of radius inch at its bottom, determine a differential equation for the height h of the water at time t. Ignore friction and contraction of water at the hole.
Read more -
Chapter 1: Problem 33 Differential Equations with Boundary-Value Problems 7
The number of field mice in a certain pasture is given by the function 200 10t, where time t is measured in years. Determine a differential equation governing a population of owls that feed on the mice if the rate at which the owl population grows is proportional to the difference between the number of owls at time t and number of field mice at time t.
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Chapter 1: Problem 34 Differential Equations with Boundary-Value Problems 7
Suppose that dAdt 0.0004332 A(t) represents a mathematical model for the radioactive decay of radium-226, where A(t) is the amount of radium (measured in grams) remaining at time t (measured in years). How much of the radium sample remains at the time t when the sample is decaying at a rate of 0.002 gram per year?
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