Notes on Diffy Qs: Differential Equations for Engineers: Errata

This page lists major problems in the various editions. Simple typos, misspellings, and such are not listed here. Also not listed are things that were correct but simply deserved improvement. Of course, older editions suffer from recently discovered errata as well.

July 16th 2010 edition:

No known errata.

July 1st 2010 edition:

No known errata.

May 18th 2010 edition:

No known errata.

April 28th 2010 edition:

On page 113, when writing the solution in terms of amplitudes and phase shifts, α₁ and α₂ were swapped.
Exercise 118 on page 118 wrongly asked for m₁ instead of m₂ which is the unknown.
On page 198 the derivation of the change of variables had a² where there should just be a in $$\frac{d}{dt}$$. Thanks to Shawn White.

April 13th 2010 edition:

On page 194 in April 13th version (or 193 in all others), the sentences about the conditions for T(0)=0 and T'(0)=0 were swapped. The final solution is OK, it is only the one sentence together with the choice of T_n (in case of the solution of z).
On page 174 in April 13th version (173 in others) the formula near the bottom should have a -4. That is, f(t) is $$\frac{1}{2} + \sum_{{n=1\\n~\text{odd}}}^\infty \frac{-4}{\pi^2 n^2} \, \cos (n \pi t)$$. Thanks to Sean Robinson.
On page 207 in April 13th version (206 in others) The formula $u_n(x,0) = X_n(x)Y_n(0) = \sin \left( \frac{n \pi}{n} x \right)$ should be $u_n(x,0) = X_n(x)Y_n(0) = \sin \left( \frac{n \pi}{w} x \right)$ (note the denominator). Thanks to Sean Robinson.

April 6th 2010 edition:

On bottom of page 173, there is the constant term of 1/4 missing in the Fourier series for the solution x.
On page 237, Heaviside function definition is off it should be 1 for all t ≥ 0.
On page 239 near middle of page, there is an $e^{-2s}$ that should be $e^{-3s}$ though the inverse transform was then done correctly. Thanks to Sean Raleigh and Jessica Robinson for spotting this one.
On page 240, when applying the Laplace transform to solve an integral equation, there was a missing equals sign and x(t) was mistakenly called f(t)
On page 241, exercises 6.2.3, 6.2.4, and 6.2.5 were trivial as given (x=0 is the solution). What I meant was to make the initial conditions arbitrary, i.e. x(0)=a and x'(0)=b. Thanks to Sean Raleigh and Jessica Robinson for spotting this one.
On page 242, exercise 6.3.2, there should be no ω₀, it should just be ω. Thanks to Sean Raleigh and Jessica Robinson for spotting this one.

March 7th 2010 edition:

Exercise 4.1.4 on page 145 had forgotten applications of chain rule in the computation. The final result is not changed. Thanks to Michael Angelini for pointing this out.
On top of page 146, the eigenfunctions given should be cos kt, not sin kt.
On page 163 top, in the definition of piecewise continuous the condition that one sided limits exist at all endpoints was written wrong, as k already referred to the number of points.
On page 168 of course it is the cosine terms disappear for an odd extension, not the sine terms.
On page 170 on the top, there is a plus sign missing in the series after $\frac{a_0}{2}$ .
On page 212 we mistakenly said that eigenvalues are lambdas with no nontrivial solutions, of course it is just the opposite (thanks to Sean Raleigh / Jessica Robinson).
On page 217, the integral that is the inner product of the sine with itself should not have f(t) in it, the one that equals $\frac{\pi}{4}$ (thanks again to Sean Raleigh / Jessica Robinson).
On page 222, Newton's second came out all jumbled up. Of course it is "force equals mass times acceleration" (and again: Sean Raleigh / Jessica Robinson).

February 17th 2010 edition:

Exercise 2.4.5 is not possible. There is a typo in the frequencies. If you replace 0.8 with 1.1 and 0.39 with 0.8 the exercise will be possible. Also what was not noted is that you always get two possibilities for the mass in the formulas.
On page 54 about mid page, the formula for the complex roots is slightly off. If we want to take out an i, then we must negate what's under the square root sign (the discriminant). That is, the roots should be: $\frac{-b}{2a} \pm \frac{\sqrt{4ac-b^2}}{2a}$ .
On page 60, in the solution corresponding to complex roots of multiplicity k, the x^k should be x^k-1.
On page 60, the factorization and completing the square of the characteristic equation in example 2.3.5 has two typos in it, even though the roots are given correctly. The procedure should proceed as (r²-2r+2)²=0 and ((r-1)²+1)²=0.
Theorem 4.1.2 talks about p and q which do not appear in the theorem (they do appear in the more complicated version from chapter 5). Thanks to Sean Raleigh and Jessica Robinson for this.
IODE Project III has changed (there was always some confusion about what was Project III) and now fits more after section 2.3, rather than before 2.6.
On page 75 on top, there is a typo in the expression for u₁', the x should be an argument to the cosine.
In example 2.6.1, when stating the parameters, it should say m=0.5 (the example is computed correctly, so no other changes needed).

February 8th 2010 edition:

Example 1.3.4 has a mistake in the integration of the left hand side. The solution should end up being $y=\frac{6}{x^2+2C}$ . Thanks to Leonardo Gomes for noticing this.
Example 2.4.1 (page 65) has a missing 2 in the derivation. Thus B is off, and so is C and gamma. Figure 2.2 is therefore also not correct. Also the discussion below the example is off by ω₀ (of course x'(0) = ω₀ B).
Example on bottom of page 72 and top of page 73 is off. The particular solution should have 1/6 not a 1/4.
The example on page 73 has a y missing in the equation, it should of course be y''-6y'+9y=e^3x.
In Example 6.1.7 (page 234), the inverse transform is off by an extra 1/2.
In Example 6.3.3 (page 243), the convolution is not computed correctly.
Figure on page 37 (bottom of page) has y=5 and y=0, when it should be x=5 and x=0. Thanks to Jeff Winegar for pointing it out.
Exercise 2.3.4 is hard to read as "equation" refers to the ODE (it is correct, but can be hard to understand).

January 10th 2010 edition:

Mildly ordered by severity:

Example on page 34 uses the wrong integrating factor. It is off by a constant. The correct integrating factor would be x^-4e^-4x+4. Thanks to Sean Raleigh and Jessica Robinson for pointing this out.
Example 1.1.3 on page 15 has the wrong constant in front when we solve for |y|. It should be $\frac{1}{|k|} e^{-C'k}$ . It doesn't (of course) change the final solution as C' was arbitrary.
Example 1.3.3 had a minus sign error in the derivation, though it does not affect the final answer (and is not technically wrong if we allow complex logarithms...). Also first we implicitly give time in seconds and then we use minutes (again computations were OK).
Example 1.4.1 was missing a minus sign in the solution. It the solution should be y=e^x-x²-2e^-x². Thanks to Ian Simon for pointing it out.
Example on page 103 has the signs wrong. The expression for x₁ in the middle of the page should have e^tcos(t) -i e^tsin(t). The rest of the example also needs to have the signs fixed. Thanks to Sean Raleigh and Jessica Robinson for noticing.
Exercise 1.1.7 has a typo making the solution very hard (without a computer algebra system). The right hand side should be a $\frac{1}{y+1}$ not $\frac{1}{y^2+1}$ .
Exercise 1.3.7 was impossible to solve as stated. Ignore this exercise (due to a copy/paste error, the equation happens to be the same as in 1.3.6, but with an impossible initial condition). It will be replaced with a different exercise in the next version of the notes.
There was a forgotten initial condition in Exercise 0.2.9 (x'(0)=0). The exercise is possible as is, but it was not what was intended.
In exercise 1.4.9, in part b, the target concentration should be 0.001, not 0.01 (obviously it can't be 0.01)
Exercise 1.5.6 is way too difficult, ignore. It will be replaced.
Exercise 1.5.1 was actually worked out as an example. It will be replaced.
Caption on Figure 1 gives the wrong equation.
Captions on Figures 1.7-1.10 have an extraneous negative sign, replace -0.1 by 0.1. The vertical scale on 1.6-1.10 is improperly rounded.

November 25th 2009 edition:

Examples on the first page of section 2.1 are wrong. The k in the equations should be k². Thanks to Thomas Wicklund for noticing.

October 23rd 2009 edition:

Exercise 2.1.8 has a missing minus sign. It should look like:
Exercise 2.1.8: Suppose $y_1$ is a solution to $y''+p(x)y'+q(x)y=0$ . Show that
$y_2(x)=y_1(x)\int \frac{e^{-\int p(x)dx}}{(y_1(x))^2} dx$
is also a solution.

September 29th 2009 edition:

Exercise 4.6.4 on page 188 is not correct. At least not what was intended. The side condition should be u(x,0) = 3 cos x + cos 3x.

May 12th 2009 edition:

Example 1.4.2 is wrong. There was an error with integration and there is a factor of 2 missing in the 5th line of the computation on page 30. Here is an updated page 29 and 30. Thanks to Thomas Wicklund for spotting this error.