# MuPAD

We first define a differential equation by defining it to be
an equation in the **ode** domain.

>> de:=ode(y'(x)+4*y(x)=exp(3*x),y(x)); ode(4 y(x) + diff(y(x), x) = exp(3 x), y(x))

Note that MuPAD supports the dash notation for the derivative.

>> solve(de); { 3 } { exp(x) } { ------- + C1 exp(-4 x) } { 7 }We can, of course, solve equations with initial conditions:

>> de2:=ode({y''(x)-2*y'(x)+3*y(x)=0,y(0)=1,y'(0)=2},y(x)); ode({D(y)(0) = 2, y(0) = 1, 3 y(x) - 2 diff(y(x), x) + diff(y(x), x, x) = 0 }, y(x)) >> solve(de2); { 1/2 1/2 } { 1/2 exp(x) 2 sin(x 2 ) } { exp(x) cos(x 2 ) + ----------------------- } { 2 }MuPAD will solve linear differential equations with little fuss, and also a large class of non-linear differential equations.

Recurrence relations are solved similarly; by defining an
equation to be of type **rec**, and applying the
command **solve**.

>> rr:=rec(a(n)=a(n-1)+3*a(n-2),a(n)); rec(a(n) = a(n - 1) + 3 a(n - 2), a(n)) >> solve(rr); { / 1/2 \n / 1/2 \n } { | 13 | | 13 | } { a1 | ----- + 1/2 | + a2 | 1/2 - ----- | } { \ 2 / \ 2 / }

If we include initial conditions, MuPAD will return the exact solution.

>> rr2:=rec(a(n)=a(n-1)+3*a(n-2),a(n),{a(0)=1,a(1)=3}); rec(a(n) = a(n - 1) + 3 a(n - 2), a(n), {a(0) = 1, a(1) = 3}) >> solve(rr2); { / 1/2 \ / 1/2 \n / 1/2 \ / 1/2 \n } { | 5 13 | | 13 | | 5 13 | | 13 | } { | 1/2 - ------- | | 1/2 - ----- | + | ------- + 1/2 | | ----- + 1/2 | } { \ 26 / \ 2 / \ 26 / \ 2 / }In the case of a non-homogeneous equation, MuPAD returns the particular solution.

>> rr3:=rec(a(n) = a(n - 1) + 3*a(n - 2)+n^2, a(n)); 2 rec(a(n) = a(n - 1) + 3 a(n - 2) + n , a(n)) >> solve(rr3); { 2 } { 14 n n } { - ---- - -- - 59/27 } { 9 3 }

Graphical display in MuPAD is produced by means of a separate
program called **VCam**, which can be
used entirely independent of MuPAD. But let's suppose we are in the
middle of a MuPAD session (started with xmupad), and we wish to
plot a few functions. The command **plotfunc** will
allow us to plot functions of the form y=f(x), or in three
dimensions, of the form z=f(x,y).

>> plotfunc(1/(1+x^2),x=-4..4); >> plotfunc(x^3-3*x*y^2,x=-2..2,y=-2..2);

Such commands will open up a VCam window with the function displayed in it, and also another window in which it is possible to change various plot options. The output of this second command is shown in Figure 2, along with various of VCams's windows.

**Figure 2. A
Screenshot of VCam in Action**

More general plot commands are **plot2d** and
**plot3d**, by which the above plots can be
generated as follows:

>> plot2d(Mode = Curve, [u, 1/(1+u^2)], u = [-4, 4]]); >> plot3d([Mode = Surface, [u, v, exp(-sqrt(u^2+v^2))], u = [-4, 4], v = [-4, 4]]);

We see that these commands use a parametric representation of the function. This allows for the easy plotting of a large number of different functions. There are a large number of plot options, including colours of plots and backgrounds, style of curves or surfaces, position and style of axes, labels and titles. All these can be changed interactively in VCam, or given as options to the plot command.

For a more complex example, let's plot a torus, with radii 3 and 1. This can be plotted with:

>> plot3d([Mode = Surface, [(3+cos(u))*cos(v), (3+cos(u))*sin(v), sin(u)], u = [0, 2*PI], v = [0, 2*PI]]);

This will give a wireframe torus with three numbered axes—not a particularly elegant result. We can add more options to this basic command to give us a nice filled-in torus with hidden lines, nestling in a box with no numbers:

>> plot3d(Title = "A torus", Axes = Box, Labels = ["","",""], Ticks = 0, [Mode = Surface, [(3+cos(u))*cos(v), (3+cos(u))*sin(v), sin(u)], u = [0, 2*PI], v = [0, 2*PI] Style = [ColorPatches, AndMesh]]);All these options can now be changed again with VCam. We can also change the perspective; zoom in and out; modify the colour scheme; and many other things. We can draw several surfaces at once, by placing multiple surface definitions in the one

**plot3d**command. For example:

>> plot3d(Title = "Two tori", Axes = Box, Labels = ["","",""], Ticks = 0, [Mode = Surface, [(5+cos(u))*cos(v), (5+cos(u))*sin(v), sin(u)], u = [0, 2*PI], v = [0, 2*PI], Style = [ColorPatches, AndMesh] ], [Mode = Surface, [sin(u), 5+(5+cos(u))*cos(v), (5+cos(u))*sin(v)], u = [0, 2*PI], v = [0, 2*PI], Style = [ColorPatches, AndMesh] ] );The result of this is shown in Figure 3.

**Figure 3. Two
Interlocking Tori**

When dealing with such long expressions as this, it is
convenient to first use your favorite editor to create a small
file, say “**tori.mu**”, to contain the above
command, then read it into MuPAD:

>> read("tori.mu");

This obviates using MuPAD's own text editing facilities, which are very basic.

As Linux continues to play an ever increasing role in corporate data centers and institutions, ensuring the integrity and protection of these systems must be a priority. With 60% of the world's websites and an increasing share of organization's mission-critical workloads running on Linux, failing to stop malware and other advanced threats on Linux can increasingly impact an organization's reputation and bottom line.

Sponsored by Bit9

Most companies incorporate backup procedures for critical data, which can be restored quickly if a loss occurs. However, fewer companies are prepared for catastrophic system failures, in which they lose all data, the entire operating system, applications, settings, patches and more, reducing their system(s) to “bare metal.” After all, before data can be restored to a system, there must be a system to restore it to.

In this one hour webinar, learn how to enhance your existing backup strategies for better disaster recovery preparedness using Storix System Backup Administrator (SBAdmin), a highly flexible bare-metal recovery solution for UNIX and Linux systems.

Sponsored by Storix

## Trending Topics

## Free Webinar

### One Click, Universal Protection: Implementing Centralized Security Policies on Linux Systems

Join editor Bill Childers and Bit9's Paul Riegle on April 27 at 12pm Central to learn how to keep your Linux systems secure.

Free to *Linux Journal* readers.

diff -u: What's New in Kernel Development | Aug 20, 2014 |

Security Hardening with Ansible | Aug 18, 2014 |

Monitoring Android Traffic with Wireshark | Aug 14, 2014 |

IndieBox: for Gamers Who Miss Boxes! | Aug 13, 2014 |

Non-Linux FOSS: a Virtualized Cisco Infrastructure? | Aug 11, 2014 |

Linux Security Threats on the Rise | Aug 08, 2014 |

- diff -u: What's New in Kernel Development
- Security Hardening with Ansible
- NSA: Linux Journal is an "extremist forum" and its readers get flagged for extra surveillance
- New Products
- Tech Tip: Really Simple HTTP Server with Python
- Monitoring Android Traffic with Wireshark
- [<Megashare>] Watch Mrs Brown's Boys Movie Online Full Movie HD 2014
- RSS Feeds
- Raspberry Pi: the Perfect Home Server
- Returning Values from Bash Functions

## Featured Jobs

Linux Systems Administrator | Houston and Austin, Texas | Host Gator |

Senior Perl Developer | Austin, Texas | Host Gator |

Technical Support Rep | Houston and Austin, Texas | Host Gator |

UX Designer | Austin, Texas | Host Gator |

Web & UI Developer (JavaScript & j Query) | Austin, Texas | Host Gator |

## Comments

## Sellout

Mupad has been bought out by mathworks and all code is now under matlab (junk) licence.

any and all open source work is now dead.

## Thankyou for a well written a

Thankyou for a well written article. TeXmacs acts as an excellent interface to mupad. I assume that the TeXmacs screen display generated by TeX. The graphics is generated by javaview. The combination of TeXmacs and javaview greatly enhance the mupad experience.