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Errata for Handbook of Differential Equations

by Daniel Zwillinger
(Orlando, Florida: Academic Press, 1997) xxiii+801pp. ISBN 0-12-784396-5

Table of contents can be viewed here
Reviews can be viewed here.

LAST UPDATED: November 22, 2000

  1. Section 11, Fixed Point Existence Theorems, page 54
    1. The name ``Schrauder'' should be ``Schauder''
    2. The following reference should be added:
      J. SCHAUDER, ``Der Fixpunktsatz in Funktionalräumen,'' Studia Math., 2, (1930), 171-180.

    (Thanks to G. Friesecke for these corrections.)

  2. Section 27, Canonical Forms, page 118, reference number 2 is now
    Bateman, H. Partial Differential Equations of Mathematical Physics, Dover Publications, New York, 1944.
    Which is incorrect. The reference should have been
    Bateman, H. Differential Equations, Longmans, Green and Co., New York, 1926, pages 75-79.

    (Thanks to Ali Nejadmalayeri for this correction.)

  3. Section 44.1.3, Look-Up Technique, page 172, last equation before section 44.2, presently has
    $ y^{(m)} =axy^{-m/2}$
    This is incorrect, it should have been
    $ y^{(m)} =ayx^{-m/2}$

    (Thanks to Flavio Noca for this correction.)

  4. Section 79, Integrating Functions, page 326, note number 10, the following should be added:
    The general solution to $ u_x=yu_y$ is $ u=f(x+\log y)$, where $ f$ is an arbitrary function.

    (Thanks to Alain Moussiaux for this observation.)

  5. Section 80, Interchanging Dependent and Independent Variables, page 327, note number 2, the reference to Bender and Orszag should be section 1.5, not 1.6.

    (Thanks to James Dare for this observation.)

  6. Section 85, Reduction of order, page 354, note number 2, presently contains
    More generally, if $ \{z_1(x),\dots,z_p(x)\}$ are linearly independent solutions of equation (85.6), then the substitution

    $\displaystyle y(x)= \begin{bmatrix} z_1 & \dots & z_p & v \\ z_1' & \dots & z... ...\vdots & \vdots \\ z_1^{(p)} & \dots & z_p^{(p)} & v^{(p)} \\ \end{bmatrix}$

    reduces equation (85.7) to a linear ordinary differential equation of order $ n-p$ for $ v(x)$.
    This should be changed to
    More generally, if $ \{z_1(x),\dots,z_p(x)\}$ are linearly independent solutions of equation (85.6), then the substitution

    $\displaystyle y(x)= \begin{bmatrix} z_1 & \dots & z_p & z \\ z_1' & \dots & z... ...\vdots \\ z_1^{(p)} & \dots & z_p^{(p)} & z^{(p)} \\ \end{bmatrix} \phi(x) $

    where $ \phi(x)$ need not be specified, reduces equation (85.6) to a linear ordinary differential equation of order $ n-p$ for $ y(x)$.

    Here $ y(x)$ can be written in the form

    $\displaystyle y(x)=A(x)z^{(p)} +B(x)z^{(p-1)} +\dots, \qquad A(x)\ne 0 $

    and its derivatives have the form

    $\displaystyle y'(x)=A(x)z^{(p+1)} +\dots, \qquad y''(x)=A(x)z^{(p+2)} +\dots, $

    These equations can be used to eliminate $ \{z^{(p)}, \dots, z^{(n)}\}$ and (85.6) will take the form

    $\displaystyle b_0y^{(n-p)}+ \dots+b_{n-p}y+ V=0 $

    where $ V$ is linear in the $ \{z,z',\dots,z^{(p-1)}\}$

    (Thanks to Unal Goktas for this correction.)

  7. Section 106, Inverse Scattering, page 416, the Applicable to statement should have at the end
    having the form of (106.2)

    (Thanks to G. Friesecke for this observation.)

  8. Section 93, Inverse Scattering, page 373, the last line contains the equation
    $ L[y]=y''+a(x)y'+b(x)=f(x)$
    Which is incorrect. This should have been (note the missing $ y$)
    $ L[y]=y''+a(x)y'+b(x)y=f(x)$

    (Thanks to Young Kim for this correction.)

  9. Section 118, Chaplygin's Method, page 465, equations (118.5) and (118.6) and the surrounding text are now
    Then define $ u_1(x)$ to be the solution of

    $\displaystyle y' = M(x)y+N(x), \qquad y(x_0)=y_0. \qquad\qquad \qquad\qquad (118.5)$    

    and define $ v_1(x)$ to be the solution of

    $\displaystyle y' = {\widehat M}(x)y+ {\widehat N}(x),\qquad y(x_0) = y_0. \qquad\qquad \qquad\qquad (118.6)$    

    Which is incorrect. This should have been (note that the definitions have been switched):
    Then define $ v_1(x)$ to be the solution of

    $\displaystyle y' = M(x)y+N(x), \qquad y(x_0)=y_0. \qquad\qquad \qquad\qquad (118.5)$    

    and define $ u_1(x)$ to be the solution of

    $\displaystyle y' = {\widehat M}(x)y+ {\widehat N}(x),\qquad y(x_0) = y_0. \qquad\qquad \qquad\qquad (118.6)$    

    (Thanks to Bruno Van der Bossche for these corrections.)

  10. Section 145, Picard Iteration, page 561, note number one, the following should be added:
    However, the successive approximations are guaranteed to converge to the true solution for all $ x$ sufficiently close to zero provided $ f$ is a continuously differentiable function.

    (Thanks to G. Friesecke for this observation.)

  11. Section 148, Soliton-Type Solutions, pages 567-569
    1. In equation (148.3) the term $ cv_{\zeta}$ should be $ -cv_{\zeta}$.
    2. In equation (148.4) the term $ (v_{\zeta})^2$ should be $ \frac{1}{2}(v_{\zeta})^2$.
    3. An additional note should be added on page 569 to state
      With the standard choice of $ A=B=0$, the solution to (148.4) can be solved in terms of elementary functions:

      $\displaystyle v(x)=\frac{3c}{\sigma}\left(\mathop{\textrm{sech}}\left(\frac{\sqrt{c}x}{2}\right)\right)^2 $

    (Thanks to G. Friesecke for these corrections.)

  12. Section 172, Pseudospectral Method, page 772 presently has:

    $\displaystyle \left. \frac{\partial u}{\partial x}\right\vert _{x=x_k} \simeq \frac{1}{3h}(u_{k+1}-u_{k-1})- \frac{1}{6h}(u_{k+2}-u_{k-2}). $

    and

    $\displaystyle \left. \frac{\partial u}{\partial x}\right\vert _{x=x_k} \simeq \... ...k+1}-u_{k-1})- \frac{1}{3h} (u_{k+2}-u_{k-2})+ \frac{1}{30h}(u_{k+3}-u_{k-3}). $

    and

    $\displaystyle \left. \frac{\partial u}{\partial x}\right\vert _{x=x_k}= \sum_{j=1}^{\infty} \frac{2(-1)^{j+1} }{jh}(u_{k+j}-u_{k-j}). $

    Which are all incorrect. They should have been:

    $\displaystyle \left. \frac{\partial u}{\partial x}\right\vert _{x=x_k} \simeq \frac{2}{ 3h}(u_{k+1}-u_{k-1})- \frac{1}{12h}(u_{k+2}-u_{k-2}). $

    and

    $\displaystyle \left. \frac{\partial u}{\partial x}\right\vert _{x=x_k} \simeq \... ...1}-u_{k-1})- \frac{3}{20h} (u_{k+2}-u_{k-2})+ \frac{1}{60h} (u_{k+3}-u_{k-3}). $

    and

    $\displaystyle \left. \frac{\partial u}{\partial x}\right\vert _{x=x_k}= \sum_{j=1}^{\infty} \frac{(-1)^{j+1} }{jh}(u_{k+j}-u_{k-j}). $

    (Thanks to Didier Clamond for these corrections.)


Errata for Handbook of Differential Equations