There is a collection of Java Applets that allow interactive work with differential equations at the Interactive Math Programs website. For example, the applet program "DIRFLD" plots direction fields; select "Print" from the "File" icon in your web browser to print a direction field produced by "DIRFLD".

The computer tools Maple and Matlab are available online from the Math Department MyMath Math Portal. Online information about Matlab is given at the Matlab information website. Some tutorial information about Maple is given at this Maple tutorial website, with more details given at this second Maple tutorial website. Here are some Maple and Matlab example commands for working with differential equations.

• Assume you want to solve y' = exp(-t) - 2y, given y(1) = 1.
  • MAPLE
    • To plot a direction field that includes a solution plot use:
      with(DEtools):
      DEplot({ diff(y(t),t) = exp(-t) - 2*y(t) },y(t),t=-2..3,[[y(1)=1]]);
    • To solve the equation symbolically use:
      dsolve( { diff(y(t),t) = exp(-t) - 2*y(t), y(1) = 1 }, y(t) );
  • MATLAB
    • To plot a direction field use:
      [T,Y] = meshgrid( -2:.2:3, -1:.2:2 );
      S = exp(-T) - 2*Y; L = sqrt(1+S.^2);
      quiver( T, Y, 1./L, S./L, 0.5 );
      axis equal tight
      title('y'' = exp(-t) - 2*y(t) Dir. Field')
      
    • To solve the equation symbolically and plot the solution use:
      sol = dsolve('Dy = exp(-t) - 2*y, y(1) = 1','t'); pretty(sol)
      ezplot( sol, [-2, 3] )
    • To solve the equation numerically and plot the solution use:
      f = inline('exp(-t) - 2*y', 'y','t'); ode45( f, [0, 3], 1 )
• In MAPLE:
  • to produce implicit equation plots for y^2/2 + exp(t)(1-2) = C, use:
    e:=y^2/2 + exp(t)*(t-1): with(plots):
    implicitplot({e=1,e=2,e=3},t=-2..2,y=-4..4);
  • to produce the first 9 terms of the series solution for cos(x)y''+xy'-2y=0, near x = 0, with y(0) = 1, y'(0) = 2, use:
    Order := 9;
    eqn := cos(x)*diff(diff(y(x),x),x) + x*diff(y(x),x) - 2*y(x) = 0;
    dsolve({eqn, y(0) = 1, D(y)(0) = 2}, y(x), type = series );
  • to plot a phase portrait for x'= 3x-2y, y'= 2x-2y, and two solutions with respective initial values [x(0)=2,y(0)=2] and [x(0)=-2,y(0)=-2], use:
    with(DEtools):
    phaseportrait([D(x)(t)=3*x(t)-2*y(t),D(y)(t)=2*x(t)-2*y(t)],[x(t),y(t)],t=-1..1,[[x(0)=2,y(0)=2],[x(0)=-2,y(0)=-2]]);