Differential Equations

The following worksheets illustrate the use of Maple on topics that typically occur in an elementary course on differential equations.

We anticipate that the worksheets may be modified and new ones added from time to time. The date at the beginning of the worksheet indicates when that particular worksheet was last modified.

• The .ms files should be downloaded and then may be incorporated in a Maple session.
• The .ps files are Postscript files, they will be displayed in an external viewer if available.

• An example of a two-dimensional competing species problem. It illustrates a nonlinear autonomous system and its linear approximation near a critical point.
  • competition.ps
    
  • competition.ms

• Three examples of direction fields and numerically calculated solution curves for scalar first order differential equations.
  • dirfields.ps
    
  • dirfields.ms

• Some examples of finding the eigenvalues and eigenvectors of a matrix.
  • eigenvals.ps
    
  • eigenvals.ms

• Five examples of the use of dsolve to find exact symbolic solutions. The examples include first order, second order, and fourth order equations, and a system of two first order equations. Some examples include initial conditions.
  • exactsols01.ps
    
  • exactsols01.ms

• An example of a first order linear equation; its direction field, numerical solution, and exact symbolic solution are obtained.
  • firstorder.ps
    
  • firstorder.ms

• An example of a Fourier series, including plots as a function of x and as a function of n, the number of terms.
  • fourier.ps
    
  • fourier.ms

• Two examples of Fourier sine and cosine series, including plots of partial sums, and in one case, plots of the limiting function.
  • fourier02.ps
    
  • fourier02.ms

• Separation of variables solution of the heat conduction equation with zero boundary conditions at each end of a finite interval. Examples show plots of u versus x, u versus t, animations of these plots, and a three-dimensional plot of u versus x and t.
  • heat.ps
    
  • heat.ms

• Exact solution of a two dimensional linear system, showing time plots and parametric plot (phase trajectory). The system has a stable spiral point; the same system is considered numerically in linsys03.
  • linsys01.ps
    
  • linsys01.ms

• Two dimensional linear systems, showing direction fields, phase portraits, time responses, and three dimensional trajectories. Illustrates nodes, saddle points, and spiral points.
  • linsys02.ps
    
  • linsys02.ms

• The same system as in linsys01 is investigated numerically, producing a phase plot, plots of x and y versus t, and a three dimensional plot in the txy space.
  • linsys03.ps
    
  • linsys03.ms

• Shows the effect of damping on the response of a linear spring-mass system, using animation and a three dimensional plot.
  • vibrations01.ps
    
  • vibrations01.ms

• Shows the effect of the forcing frequency on the response of a linear vibrating system. Makes use of animation and a three dimensional plot.
  • vibrations02.ps
    
  • vibrations02.ms