Andrew Knyazev's comments

A Course in Ordinary Differential Equations (typos and comments)

by Randall J. Swift and Stephen A. Wirkus Chapman Hall/CRC; 1 edition 2006 ISBN: 1584884762

• Only Examples 1.2.1 and 1.2.2 include the description of the domain for the solution. In many other examples in Chapter 1 this is missing and must be added.
• Example 1.2.3: a) a discussion of why one can divide by x is missing; b) a discussion of why one can divide by y^2-3 is missing.
• Example 1.2.4: the solution has a singularity at some point on the interval [2,3], so an explanation is needed why the initial condition at the point x=2 determines the constant at the point x=3.
• Example 1.2.9: a reasonably simple explicit solution for y can be obtained with a bit more work and perseverance.
• Equations (1.12) and (1.13) are equivalent only if x=const is not a solution of (1.2). In (1.13), x=const can never be a solution since (1.13) has dx in the denominator.
• Computer Code 1.9: Numerically solving a differential equation with built-in commands and plotting this solution. The m-file is called Example5.m while the call is [x1,y1]=ode45('Example4',[2,5],1); It is also a common practice in MATLAB to use the same m-file name as the function name (unless there is more than one function in the file). So it would be more appropriate to use function f=Example5(xn,yn)
• Computer Code 2.5:
  • The following two lines

    %Note that function RK4step is contained in RK4.m!!
    %Function RK4.m ends with "% end of m-file RK4.m"
    

    should be removed.
  • It should be axis([-7,7,-0.5,2.5]) not axis[-7,7,-0.5,2.5]
  • axis wrong?

      close all; clear all;
      Example3=inline('sin(xn.^2)','xn','yn') %inline function
      [x1,y1]=RK4(Example3,[0,1],1,1.0);
      [x2,y2]=RK4(Example3,[0,1],1,0.5);
      [x3,y3]=RK4(Example3,[0,1],1,0.1);
      plot(x1,y1,'r:');
      hold on
      plot(x2,y2,'b-')
      plot(x3,y3,'g-.')
      axis([-7,7,-0.5,2.5])
      xlabel('x');ylabel('y');
      title('dy/dx=sin(x2); dotted is h=1.0, dashed is h=0.5,dot-dash is h=0.1');
      hold off
    

    The figure does not look good.
• Computer Code 3.4 small suggestion:
  The book says: Note that if we wanted to use MATLAB’s ode45, we would simply type MATLAB [x1,y1]=ode45(’Example1’,[x0,xf],y0); instead of the above two lines that use RK4. My comment: this is not technically correct since in the second line one should use [x1,y1]=ode45(’Example2’,[x0,xf],y0); Small suggestion to fix it:

  %Computer Code 3.4: Numerically solving a higher-order
  %differential equation with fourth-order Runge-Kutta method and
  %built-in commands; plotting this solution
  %Example1.m
  function f=Example1(xn,yn)
  %
  %The original ode is 2*x*y’’(x) +x∧2*y’(x)+3*x∧3*y(x)=0
  %The system of first order equations is
  %u1’=u2
  %u2’=(-x/2)*u2-((3*x∧2)/2)*u1
  %
  %We let yn(1)=u1, yn(2)=u2
  %
  f= [yn(2); (-xn/2)*yn(2)-((3*xn∧2)/2)*yn(1)];
  %Example2.m
  function f=Example2(xn,yn)
  %The original ode is y∧(4)(x)+ x∧2*y’(x)+y(x)=cos(x)
  %The system of first order equations is
  %u1’=u2
  %u2’=u3
  %u3’=u4
  %u4’=-x∧2*u2-u1+cos(x)
  %
  %We let yn(1)=u1, yn(2)=u2, yn(3)=u3, yn(4)=u4
  f= [yn(2); yn(3); yn(4); -xn∧2*yn(2)-yn(1)+cos(xn)];
  %
    x0=1; xf=5;
    y0=[2, 0];
    [x1,y1]=ode45(’Example1’,[x0,xf],y0);
    %Alternatively: [x1,y1]=RK4(’Example1’,[x0,xf],y0,.05);
    subplot(211),plot(x1,y1(:,1))
    xlabel(’x’); ylabel(’y(1)’);
    subplot(212),plot(x1,y1(:,2))
    xlabel(’x’); ylabel(’y(2)’);
    t=length(x1);
    y1(t,:) %shows the entries of y1 corresponding to the
    figure
    x0=0; xf=5;
    y0=[1, 0, 0, 1];
    [x1,y1]=ode45(’Example2’,[x0,xf],y0);
    %Alternatively:[x2,y2]=RK4(’Example2’,[x0,xf],y0,.05);
    plot(x2,y2)
    legend(’y(1)’,’y(2)’,’y(3)’,’y(4)’)
    xlabel(’x’); ylabel(’y(1), y(2)’);
  

• Computer Code 4.1 small improvement

  Computer Code 4.1: Finding all roots of a polynomial
  MATLAB, Maple, Mathematica
  MATLAB
   >> p=[1 -4 1 6] %these are the coeff of the poly
  
  Why not
    p=[1 -4 1 6] %these are the coefficients of the polynomial
  or shorter
    p=[1 -4 1 6] %the coefficients of the polynomial
  

  The same in CC 4.2.
• Computer Code 5.4: Calculating eigenvalues and eigenvectors: "students often prefer to have “nicer looking” eigenvectors and eigenvalues." Eigenvalues cannot be chosen nicer looking
• Formal definitions of the autonomous equation and the equilibrium solutions on p. 102-103 would be helpful.
• On p. 103 just above (2.4) it is not necesserely a "growth" equation, for negative r.
• Figure 2.9 and below. The direction field does not appear to actually be in the y'-y coordinates, but should rather be in the x-y coordinates.
• A definition of an asymptotically stable solution, judged by its name, should be weaker compared to the definition of a stable solution, but in Definition 2.1 it is the opposite.
• Why x>0 is required in the formula in the middle of p.105?
• On p. 125, the MATLAB code Euler1.m uses dynamic memory allocation for xout and yout, which makes it unnecessary slow.
• On p. 177, formula (3.15) is WRONG, because the formula for w above is wrong. Is is repeated multiple times later in this wrong form, up to problem 22. The correct formula must have the term 1/f^2 OUTSIDE of the exponent, instead of the constant c, which is irrelevant.