Engineering Differential Equations: Theory and Applications, Springer 2010

This page contains supplementary material for the book, Engineering Differential Equations: Theory and Applications, by Bill Goodwine. 2010, Springer.

Movies

Chapter 11 considers solutions to partial differential equations.

The One-Dimensional Wave Equation

Section 11.1 considers the one-dimensional wave equation.

Some movies:

• String of length 3 plucked at L=1 (modeling a guitar):
  • The first (fundamental) mode of the plucked string
  • The first five modes of the plucked string plotted separately
  • The sum of the first five modes of the plucked string
  • The sum of the first fifty modes of the plucked string
• String of length 3 that is impacted near L=1 (modeling a piano):
  • The sum of the first ten modes of the impacted string
  • The sum of the first fifty modes of the impacted string

The One-Dimensional Heat Conduction Equation

Section 11.3 considers the one-dimensional heat conduction equation.

Some movies:

• The solution to the heat equation with inhomogeneous boundary conditions
• The solution to the heat equation with an insulated end

The Two-Dimensional Heat Equation

Movie:

• Solution to the heat conduction equation in two dimensions

Vibrating Membranes

Partial differential equations describing vibrating membranes in rectangular and polar coordinates are considered in Section 11.5.

Section 11.5.1 presents the two-dimensional wave equation in rectangular coordinates.

Some movies:

• 1-1 mode for rectangular drum
• 2-1 mode for rectangular drum
• 2-2 mode for rectangular drum
• Rectangular drum impacted near a corner

Sections 11.5.2-11.5.4 presents the wave equation in polar coordinates. The prototypical example would be a vibrating drum head.

Some movies:

• 1-0 mode for a circular drum
• 1-1 mode for a circular drum
• 1-2 mode for a circular drum
• 1-3 mode for a circular drum
• 2-0 mode for a circular drum
• 2-1 mode for a circular drum
• 4-3 mode for a circular drum
• Circular drum that is impacted by a drum stick
• Another drum impact movie
• Cool youtube movie of slow motion impact of drum head and cymbal

Errata

Thank you to the many students who, over the years, have found mistakes and pointed them out to me. Also, a special thanks to Mike Stanisic and Jim Schmiedeler who also passed along many corrections to me.

Chapter 1

• p. 58, line -3, Equation 2.4: the c in the denominator should be the specific heat.

Chapter 2

• p. 61, second paragraph of Section 2.2, line 2: "and" should be "is".
• p. 68, line 3: the constants in the particular solution should be the same, c_1.
• p. 89, line -7: the right hand side of the equation below equation 2.35 should be a function of t, not x.

Chapter 3

• p. 97, line -3: there is an extra closing parenthesis in the equation for the Wronskian.
• p. 105, line 12: both exponents should be $-\omega _{n}$.
• p. 117, Figure 3.5: the figure plots the solution to ${\ddot {x}}+{\dot {x}}+10x=0$ and not $2{\ddot {x}}+{\dot {x}}+10x=0$ as stated. It really doesn't change the qualitative aspects of the problem, however.

Chapter 4

• p. 124, line 2: the word damped should be italicized.
• p. 125, line 5: one side of the second equation should be multiplied by -1.
• p. 125, line 8: one side of the second equation should be multiplied by -1.
• p. 125, line -4: the coefficient of the x term in the differential equation should be squared.
• p. 128, Equation 4.8: the numerator of the coefficient of the sine term should be the initial velocity.
• p. 138, line 6 (large boxed equation): the first zeta in the parentheses in the coefficient of the second exponential on the second line should be negative.
• p. 139, Equation 4.14: the second term should contain the derivative of x, not x.
• p. 142, Figure 4.16: the figure is incorrect in that the square root was not used to plot the curves.
• p. 156, Exercise 4.22: at this point in the book we don't know how to find the homogeneous solutions for these two systems.
• p. 158, Exercise 4.26, Figure 4.32: the variable x is not the distance along the beam, but rather the displacement at the end.
• p. 159, Exercise 4.27, Figure 4.33: the angle is incorrectly marked as the angular velocity.

Chapter 5

• p. 163, line 6: the first (n+1) term should be (n+2).
• p. 164, Equation 5.5: the first coefficient in the second series should be (n+2) not (n+1).
• p. 167, lines -4,-3 and -2: the expressions for the derivatives of the function are wrong.
• p. 170, line 6: at the end of the line "Table 5.2" should be "Table 5.1."
• p. 172, line -5: there should be a negative sign on the right hand side of the equation.
• p. 174, Equation 5.14: the first term in the numerator on the right hand side should be multiplied by 20.
• p. 181, line 16: the two terms on the right-hand side of the equation are switched, leading to the wrong answer for $a_{2}$. This wrong number then propagates to lines -1 and -6 on the same page.
• p. 182, line 14: in the first series for the second derivative, the exponent should be (n-2) not (n-1).
• p. 185, line 2: the recursion relation has two mistakes. The left hand side should be for a_{n+2} not {n+1}. Also, the negative sign should not be there.
• p. 186, line 6: the left hand side of the recursion relation should be for a_{n+2} not {n+1}.
• p. 188, Exercise 5.1, part 2: should be "2. Is the point t=0 an ordinary point or singular point?"
• p. 189, Exercise 5.2, part 2: should be "2. Is the point t=0 an ordinary point or singular point?"
• p. 190, Exercise 5.13: the middle term in the denominator of the second term should be "2 t" not "t2".
• p. 191, Exercise 5.16: the independent variable in the equation is x, so the derivative terms should be with respect to x.

Chapter 6

• p. 195, line 1: the third term in the vector on the right hand side should have a subscript of 4, not 3.
• p. 197, line -1: the left hand side of Equation 6.7 should be xi, not the derivative of xi.
• p. 201, line 16: there should be a hat on the xi term.
• p. 205, lines -7 and -9: the coefficient of the constant before the last term in each equation should be c_n not c₁.
• p. 206, lines 4 and 7: the expressions for the eigenvectors are missing the hat.
• p. 209, line 9: the frequency should be 2 not 1.
• p. 209, line -8: the exponential term should be multiplying the second term as well.
• p. 210, line 1: same error as on p. 209 carried through.
• p. 220, line -9: the superscript for the second term in the equation should be m-3.
• p. 223, line 4: all the 2s should be 1s in the middle matrix.
• p. 238, lines -3 and -3: the -4/25 should be 4/25.
• p. 244, line -11: the first component of the vector on the far right should contain t, not tau.
• p. 245, line 8: the matrix A₇ does not have a set of n linearly-independent eigenvectors.
• p. 247, line -7: A₃ should be A₂.

Chapter 7

• p. 253, line 5: the word "of" is repeated.
• p. 256, lines -5 and -6: the frequencies in the sine and cosine terms should be the second natural frequency, not the first.
• p. 265, Equation (7.16): this equation has multiple typos. The last term in parentheses in the last line should be (1-k1-2 k3). Also the second line has misaligned parentheses.

Chapter 8

• p. 288, last line in Table 8.1: the exponent should be "-a t" and not "a t".
• p. 297, Example 8.14: the "3" multiplying the $t^{2}$ term on the right hand side is dropped half way through the example. Because it scales the whole right hand side, it scales simply scales the solution by 3. Thus the final answer on p. 299 should be multiplied by 3, as should the equation on the top of page 299. Also the two equations in the middle of p. 298 involving X(s) should have the right hand side multiplied by 3.
• p. 326, line -10: "Figure 8.8" should be "Figure 8.37".
• p. 327: the pendulum in Figure 8.36 should indicate a length, l, for the pendulum.

Chapter 9

• p. 348, line 4: the numerator for both expressions of G1 should be 5.
• p. 348, line 6: the numerator for both expressions of G1 should be 10.
• p. 349, caption to Figure 9.16: the numerator for G1 should be 4 in the numerator for G2 should be 9.
• p. 354, line 9: the word "row" should be "column".
• p. 359, peak time derivation: multiple mistakes
• p. 368, line 4: "r=10, 1, -1, and 10" should be "r=10, 1, -1, -10".
• p. 368, line -4: "Systems with left half-plane zeros are called..." should be "Systems with right half-plane zeros are called...".
• p. 392, Table 9.1, rule 1: This rule should be: "Plot the poles and zeros of G(s). There will be n_p branches and each branch if the root locus starts at one of the poles and n_z of the branches will end at the zeros (Rules 9.1 and 9.2).
• p. 404, lines 11 and 12, the coefficient $A$ is missing from both lines.
• p. 417, line -9: the arctan should be a ratio of the imaginary to real components of $G(i\omega )$.
• p. 437, exercise 9.7: the numerator in the second transfer function should be 4 times 20 = 80, not 4 to give comparable steady-state values for the responses.

Chapter 11

• p. 493, Fig. 11.4: The magnitude of the curve for each mode is slightly too large.
• p. 496, Fig. 11.6: the label for abscissa should be "x" not "t".
• p. 497, line 14: "the magnitude of the coefficients b_n" should be "the magnitude of the coefficients a_n".
• p. 503, line 7: "<x,z> and <x,z>" should be "<x,y> and <x,z>".
• p. 511, sentence after Theorem 11.1: "Figures 11.11 and 11.17" should be "Figures 11.14 and 11.17".
• p. 513, line 2: the coefficient is alpha^2, not alpha.
• p. 515, line 6: the "x" should be outside, not inside, the square root.
• p. 516, line -4: the denominator in the second integral should be 10, not L.
• p. 524, line 1: the sine function should not equal n pi, the argument to it should.
• p. 525, Equation 11.43: the denominator in both exponentials should be $L_{x}$.
• p. 525, Equation 11.44: the first term in the denominator of the coefficient in front of the integral should be $L_{x}$, not $L$.
• p. 532, line 12: on the right-hand side of the equations, one of the terms should be the steady-state solution.
• p. 534, Equation 11.51: one of the partial derivatives should be with respect to x.
• p. 534, Equation 11.52: the frequency in the sine and cosine functions should be multiplied by t.
• p. 536, line 10: "right-hand side of Equation (11.51)" should be "left-hand side of Equation (11.51)".
• p. 539, Figure 11.28: in the label on the vertical axis, the subscript on z should be "0,n" not "1,n".
• p. 542, last boxed set of equations: the expressions for the c and d coefficients should be in terms of the function g not the function f. Also, the d coefficient should have a sine function instead of cosine. Also, also, the second integral in the denominator of each equation (with the Bessel function squared) should include an r term. This error is also in the equations for the coefficients on the top of the page and on the bottom of the previous page.
• p. 558, Eq. 11.87: the entire equation should be multiplied by $c_{4}$.
• p. 559, line -4: the tension and mass per unit length appearing in the frequency term should both be the square root of those terms.
• p. 561, line 1: the second word "subsection" should be "subsections".
• p. 570, Exercise 11.2: add to the problem statement "let $m=1,b=0$ and $k=9\pi ^{2}$."
• p. 571, line 21: for Exercise 11.4, number 4, the initial condition should be for u(x,0) not the partial derivative of u with respect to t.
• p. 572, Exercise 11.9: in the "denominator" of the operators the partial sign is "squared" when it should be the x and y, respectively.

Chapter 12

• p. 580, line 16: the word "so" should be deleted near the end of the line.
• p. 580, line -14: the exact solution is cos(t), not -cos(t).
• p. 588, line 2: the middle term in the numerator should be evaluated at 2.0, not 1.5.
• p. 590, line 16: "c₂" should be "c₃".
• p. 592, lines 11 and 12: the minus sign multiplying the cubed term was dropped. Also, it would be more complete if the expression for f(x(t),t) were substituted into the final expression.
• p. 600, line 10: the error term on the right-hand side should be at two time steps back, not one.
• p. 622, line 9: the numerator in the last term in the second equation in Equation 12.38 should not be squared.
• p. 625, Exercise 12.2, number 6: the first derivative term should be a second derivative.
• p. 626, Exercise 12.3, number 6: the first derivative term should be a second derivative.

Chapter 13

• p. 662, Figure 13.24: the output Y(s) should be X(s) because that is what it is called in the text that refers to the figure.
• p. 662, line 8: the second equation on the page should have X(s) on the left hand side, not G_p(s).
• p. 664, line 1: the sine term inside the friction function should be multiplied by M.
• p. 654, line 1, Equation (13.23): the second equilibrium point should be x=2, x'=0, not x=0, x'=2.
• p. 664, line 4: the term multiplying a on the left hand side of the equation should be the reciprocal of what is printed.
• p. 664, line 8: the term multiplying b on the left hand side of the equation should be the reciprocal of what is printed. Also, there should be an M multiplying the sine function that is the argument to the friction() function and no M multiplying the sine function outside the friction() function.
• p. 666, line 5: the "0.3" in the equation should be "-0.3".
• p. 679, exercise 13.12, number 2: "G(s) =" is missing from the left side of the equation.

Appendix E

• Various FORTRAN programs have semicolons terminating lines (always in the declaration statements): E.2.0.20, E.2.0.21, E2.0.22, E.2.0.23, E2.0.24, E2.0.25, E.2.0.26. These semicolons should not be there.
• p. 727, program in E.2.0.18, for Example 1.30: the program contains the variable n which does nothing. The declaration, initialization to zero and increment inside the do loop should be removed.
• p. 733, line 19: the equation to update x(2) should not have an x(1) in it, it should be copy(1). Also, while it will not affect how the program works, x(2) should be copy(2) for consistency.
• p. 737, line 5: The program in E.2.0.31 is for Example 12.17, not Example 12.16.

Bibliography

• p. 740, lines 10 and 11: Bibliography items 39 and 40. Isaac Newton prefers to have his name spelled "Newton" not Netwon.