Errors in the Second Edition of
Handbook of Differential Equations

LAST UPDATED: April 21, 1998

NOTE THAT THERE MAY EXIST ADDITIONAL ERRORS IN THIS EDITION. PLEASE LOOK AT THE LIST OF ERRATA FOR LATER EDITIONS.

(Great appreciation is extended to Alain Moussiaux for identifying most of the errors listed below.)

1.

On page 13, the definition of the Wronskian contains

$$\begin{vmatrix} y_1 & y_2 & \dots & y_n \\ y'_1 & y'_2 & \... ... \\ y_1^{(n)} & y_2^{(n)} & \dots & y_n^{(n)} \\ \end{vmatrix}$$

This is incorrect, it should have been

$$\begin{vmatrix} y_1 & y_2 & \dots & y_n \\ y'_1 & y'_2 & \... ..._1^{(n-1)} & y_2^{(n-1)} & \dots & y_n^{(n-1)} \\ \end{vmatrix}$$

2.

On page 37, first line after equation (6.12), we now have

we find that B²-4AC=) and

This is incorrect, it should have been

we find that B²-4AC=0 and

3.

On page 41, line 2 of Special Case 2, the equation beginning ``$p_{ij}=F_{ij}(z_1,\dots$'' should have been ``$p_{ij}=f_{ij}(z_1,\dots$''.

4.

On page 48, just above the displayed equation, we now have

If we form the equations $\{ A, B, y A, y B, y_x B, \partial_x B, y \partial_x A \}$, then

This is incorrect, it should be (note the last element in the list)

If we form the equations $\{ A, B, y A, y B, y_x B, \partial_x B, y \partial_x B \}$, then

5.

On page 55, Note number 1, repeated ``a'' should be removed.

6.

On page 80, first displayed equation, we now have a ``c<sub>l</sub> '' appearing, this should have been ``c_y ''.

7.

On page 81, first theorem, line 3, we now have ``$\vert\text{\bf f}\vert\to0$'' appearing, this should have been ``$\vert\text{\bf y}\vert\to0$''.

8.

On page 82

(a)

Case (A), line 3, we now have ``$\delta$ exists'' appearing, this should have been ``$\epsilon$ exists''.

(b)

Case (C), line 2, we now have ``hatb '' appearing, this should have been ``$\text{\(\hat b\)}$''.

(c)

Case (E), line 2, we now have ``hatb '' appearing, this should have been ``$\text{\(\hat b\)}$''.

9.

On page 111, reference 3, year is now ``190'', it should have been ``1990''.

10.

On page 124

(a)

First displayed equation, we now have ``$$1-\frac{1}{2}\frac{M}{x^2}+\dots$$'', this should have been ``$$-1-\frac{1}{2}\frac{M}{x^2}+\dots$$''.

(b)

Equation (31.5), we now have ``$$\phi_{\infty} + x+\dots$$'', this should have been ``$$\phi_{\infty} - x+\dots$$''.

11.

On page 126, line -5 , delete the word ``name''.

12.

On page 127, last line before Transformation 4 is the expression ``$\dots+Bx\sin cx$'', this should have been ``$\dots+Bx\sin ax$''.

13.

On page 129, line 2, the word ``dependent'' should have been ``independent''.

14.

On page 145, line -6 , the expression ``$\{\lambda_k]\vert$'' should have been ``$\{\lambda_k\}$''.

15.

On page 157, first displayed equation, the ``z '' in the numerator on the right hand side should have been a ``x ''.

16.

On page 169, first displayed equation, the ``a '' in the second equation should not be present.

17.

On page 186, third word is ``geberal'', it should be ``general''.

18.

On page 187

(a)

line -3 , the last term in the equation is ``(-1)<sup>0</sup>[a<sub>n</sub>(x)]w '', which is incorrect. It should have been ``(-1)⁰[a_n(x)w] ''.

(b)

line -1 , the first summation is given as ``$$\sum_{k=1}^{n-1}$$'', which is incorrect. It should have been ``$$\sum_{k=0}^{n-1}$$''.

19.

On page 194, part (C), the final equation is listed as ``$\text{\bf x}'=A x+\text{\bf f}(\text{\bf x},\text{\bf h}(\text{\bf x}))$''. This should have been ``$\text{\bf x}'=A \text{\bf x}+\text{\bf f}(\text{\bf x},\text{\bf h}(\text{\bf x}))$''.

20.

On page 195, first displayed equation is

$$u(x)= \exp \! \left[ (n-1) \int^x P(t) \,dt \right] \left\... ...xp \! \left[ (1-n) \int^s P(t) \,dt \right] Q(s) \,ds \right\}.$$

This is incorrect, it should have been

$$u(x)= (1-n) \exp \! \left[ (n-1) \int^x P(t) \,dt \right] ... ...xp \! \left[ (1-n) \int^s P(t) \,dt \right] Q(s) \,ds \right\}.$$

21.

On page 224, line 3, ``to be same'' should have been ``to be the same''.

22.

On page 228, equation (54.19) and the last displayed equation on the page, the summations are ``$$\sum_{n=0}^{\infty}$$'', which are incorrect. They should have been ``$$\sum_{n=1}^{\infty}$$''.

23.

On page 234, equation (56.2) is given as

$$(1-x)\left[ y\frac{d^{2} y' }{d {x}^{2}}-\left( \frac{d {y'}}{d {x} } \right)^2 \right] +x^2{y'}^2=0,$$

This is incorrect, it should have been

$$(1-x)\left[ y'\frac{d^{2} y' }{d {x}^{2}}-\left( \frac{d {y'}}{d {x} } \right)^2 \right] +x^2{y'}^2=0,$$

24.

On page 244, first displayed equation is now

$$\frac{d^{n} P_n }{d {x}^{n}} -\frac{d^{n-1} P_{n-2} }{d {x}^... ...^{n-2}} -\cdots + (-1)^{n-1}\frac{d {P_1}}{d {x} }+(-1)^nP_0=0.$$

This is incorrect, it should have been

$$\frac{d^{n} P_n }{d {x}^{n}} -\frac{d^{n-1} P_{n-1} }{d {x}^... ...^{n-2}} -\cdots + (-1)^{n-1}\frac{d {P_1}}{d {x} }+(-1)^nP_0=0.$$

25.

On page 249, second to last displayed equation

$$L[u]=\frac{W(u_1,u_2)}{u_1}\frac{d {}}{d {x} } \! \left[ \f... ..._2)} \frac{d {}}{d {t} } \left( \frac{u}{u_1} \right) \right],$$

This is incorrect, it should have been

$$L[u]=\frac{W(u_1,u_2)}{u_1}\frac{d {}}{d {x} } \! \left[ \f... ..._2)} \frac{d {}}{d {x} } \left( \frac{u}{u_1} \right) \right],$$

26.

On page 250, note number 4, we now have

$$\left\{ \frac{d {}}{d {t} }-q(t) \right\} \left\{ \frac{d {... ...d^{2} w }{d {t}^{2}}+w\left\{ \frac{d {q}}{d {t} }-q^2 \right\}$$

This is incorrect, it should have been:

$$\left\{ \frac{d {}}{d {t} }-q(t) \right\} \left\{ \frac{d {... ...d^{2} w }{d {t}^{2}}+w\left\{ \frac{d {q}}{d {t} }-q^2 \right\}$$

27.

On page 260, equation (65.6), we presently have

$$s F(s)-f(0)+\sqrt{s} F (s) - \frac{d^{-1/2} f(0) }{d {x}^{-1/2}} +F (s)=0$$

This is incorrect, it should have been:

$$s F(s)-f(0)+\sqrt{s} F (s) - \frac{d^{-1/2} f(0) }{d {x}^{-1/2}} -2F (s)=0$$

28.

On page 274, line -6 , the ``(68.11)'' should have been ``(68.8)''.

29.

On page 287, equation (72.18), the middle expression is

$$\bigg[ \exp \! \left( -z\xi -\frac{\xi^3}{3} \right) \bigg\vert_C$$

This is incorrect, it should have been:

$$\bigg[ \exp \! \left( z\xi -\frac{\xi^3}{3} \right) \bigg\vert_C$$

30.

On page 305, second to last line on the page, the $u(x,y)=u(y)=\exp\!\left( \int^yg(z)\,dz \right)$'' should have been $u(x,y)=u(y)=\exp\!\left( -\int^yg(z)\,dz \right)$''.

31.

On page 307, note number 3, the ``$/int M_1\,dx$)'' should have been ``$/int M_1\,dx$)''.

32.

On page 327, equation (80.4) is given as

$$y(x^2-y^2-xy)\,dx + x(y^2-x^2-xz)\,dy + xy(x+y)\,dz=0.$$

This is incorrect, it should have been (note one x should have been a z )

$$y(x^2-y^2-yz)\,dx + x(y^2-x^2-xz)\,dy + xy(x+y)\,dz=0.$$

33.

On page 331

• The second to last displayed equation on the page is given as

  $$v(x)=\frac{3}{2x}+ \frac{A}{x}\int^x\frac{e^{t^2}}{t^2}\,dt+B,$$

  This is incorrect, the equation should have been

  $$v(x)=\frac{3}{2x}+ A\int^x\frac{e^{t^2}}{t^2}\,dt+B,$$

• The last displayed equation on the page is given as

  $$y(x)=\frac{3}{2} + A\int^x\frac{e^{t^2}}{t^2} \, dt+Bx.$$

  This is incorrect, the equation should have been

  $$y(x)=\frac{3}{2} + Ax\int^x\frac{e^{t^2}}{t^2} \, dt+Bx.$$

34.

On page 333, equation (82.2) is given as $$y(x)=\frac{w'(x)}{w(x)} \frac{1}{a(x)}$$. This is incorrect, it should have been (note the minus sign) $$y(x)= - \frac{w'(x)}{w(x)} \frac{1}{a(x)}$$

35.

On page 334, equation (82.6) is given as $$y(x)=\frac{w'(x)}{w(x)} e^{-x}$$. This is incorrect, it should have been (note the minus sign) $$y(x)= - \frac{w'(x)}{w(x)} e^{-x}$$

36.

On page 339

• equation 84.2 is presently: .This is incorrect, it should have been: .
• last line, the ``(84.)'' should have been ``(84.1)''.

37.

On page 358, second displayed equation, second equation is given as ``$$v_2(x) = \int \frac{-\sin x \csc x}{-1} dx = -x.$$''. This is incorrect, it should have been ``$$v_2(x) = \int \frac{ \sin x \csc x}{-1} dx =x$$.

38.

On page 372:

• Equation (94.17) is given as: This is incorrect, it should have been .
• Equations (94.18) is given as:
  $$\begin{align*} u&=-\frac{ (x-s)^2}{1+s(x-s)^2}, \\ \tag{94.18} z&= \frac{s^3}{3}+\frac{s^2}{2}(x+y)+s(xy-1)+y. \end{align*}$$
  This is incorrect, it should have been
  $$\begin{align*} u&=\frac{ (x-s)^2}{1-s(x-s)^2}, \\ \tag{94.18} z&=-\frac{4s^3}{3}-\frac{s^2}{2}(x+y)+s(xy+1)+y.\end{align*}$$
  

39.

On page 385:

• The matrices listed as: $S= \begin{pmatrix} 1 & - 1 \\ 2 & \phantom{-}1 \\ \end{pmatrix}$and $S^{-1}= \begin{pmatrix} 1/3 & - 2/3 \\ 1/3 & \phantom{-}1/3 \\ \end{pmatrix}$are incorrect, they should have been $S= \begin{pmatrix} 2 & \phantom{-}1 \\ 1 & - 1 \\ \end{pmatrix}$and $S^{-1}= \begin{pmatrix} 1/3 & \phantom{-}1/3 \\ 1/3 & - 2/3 \\ \end{pmatrix}$.
• Equation (98.4) is listed as $$\begin{split} u_1(x,t)&=\phantom{-}w_1(x,t)+ 2 w_2(x,t) = \... ...hantom{2} w_2(x,t)= - f(x-10t)+\phantom{2} g(x-7t).\end{split}$$This is incorrect, it should have been $$\begin{split} u_1(x,t)&= 2 w_1(x,t)+w_2(x,t) = 2 f(x-10t)+g... ...} w_1(x,t)-w_2(x,t) = \phantom{2} f(x-10t)-g(x-7t).\end{split}$$

40.

On page 386:

• The first displayed equation is:

  $$\begin{split} f(x)+2 g(x)&= 3\sin 2x, \\ - f(x)+\phantom{-}g(x)&=0,\end{split}$$

  This is incorrect, it should have been:

  $$\begin{split} 2 f(x)+ g(x)&= 3\sin 2x, \\ \phantom{2} f(x)- g(x)&=0,\end{split}$$

• The second displayed equation is:

  $$\begin{split} u_1(x,t) &=\phantom{-}\sin(2x-20t)+ 2 \sin(2x... ...x,t) &=- \sin(2x-20t)+ \phantom{2} \sin(2x-14t). \\ \end{split}$$

  This is incorrect, it should have been:

  $$\begin{split} u_1(x,t) &= 2 \sin(2x-20t)+ \sin(2x-14t), \\ ... ...(x,t) &= \phantom{2} \sin(2x-20t)- \sin(2x-14t). \\ \end{split}$$

41.

On page 392, first line contains ``were quasilinear''. This is incorrect, it should have been ``were linear''.

42.

On page 579, the 7th displayed equation is now

$$f''(x_0)=\frac{-f_3+4f_2-5f_1+2f_0}{h^2}+O\left( h^3 \right)$$

the quantity h³ is in error, it should be h² . That is, the formula should read

$$f''(x_0)=\frac{-f_3+4f_2-5f_1+2f_0}{h^2}+O\left( h^2 \right)$$

43.

On page 581, the first two displayed equations are:

$$\begin{align*} \frac{\partial u}{\partial x}\left.\strut\right\vert _{\text{\bf ... ...{0,0}} &= \frac{u_3-u_1}{h(\theta_1+\theta_3)} +O\!\left( h \right) \end{align*}$$
and
$$\begin{align*} \frac{\partial^{2} u}{\partial {x}^{2}}\left.\strut\right\vert _{... ...\frac{1}{\theta_2\theta_4} \right)u_0 \right] +O\!\left( h \right) \end{align*}$$

They are incorrect, they should have been (note the interchange of x and y in the first two equations, and the three needed modifications $\theta_2\to\theta_4$, $\theta_1\to\theta_3$, and $\theta_2\to\theta_4$):

$$\begin{align*} \frac{\partial u}{\partial x}\left.\strut\right\vert _{\text{\bf ... ...{0,0}} &= \frac{u_2-u_4}{h(\theta_2+\theta_4)} +O\!\left( h \right) \end{align*}$$
and
$$\begin{align*} \frac{\partial^{2} u}{\partial {x}^{2}}\left.\strut\right\vert _{... ...\frac{1}{\theta_2\theta_4} \right)u_0 \right] +O\!\left( h \right) \end{align*}$$

44.

On page 645, equation (166.2), the terms u_x and u_y should be replaced by v_x and v_y .