## Math 285 Spring 2010 - Tips on using Wolfram Alpha

Matlab, Maple and Mathematica are software packages that can solve a huge variety of mathematical problems, using both symbolic and numerical methods.
The makers of Mathematica have created a free website, called Wolfram Alpha, that allows users to ask many types of scientific and mathematical question using essentially natural language.

You might use Wolfram Alpha for checking your answers, for plotting functions, or for exploring a homework problem as you try to solve it (in this course and your other mathematics and engineering courses). You can also use the Iode software package, which is specifically adapted to courses in differential equations.

This page collects some input commands to Wolfram Alpha that are useful for our differential equations course. See the Wolfram Alpha website for lots more examples from Mathematics, Physics, Engineering, etc. See also the Wolfram Alpha Wiki.

### Sample input to Wolfram Alpha

#### Differential equations

Solving a differential equation:
`solve y'=2y`
`solve y'=3y^2`
`solve y''+(3^2)y=0`

Solving a differential equation with initial condition:
`solve y'=2y, y(0)=3`

#### Graphing functions

Plotting a function: `plot 3e^(2x), x=-1 to x=1`
Plotting another function: `plot sin(3 pi x) with x=-1 to x=1`
Plotting a function of two variables: `plot x^2+y^2, x from -1 to 1 and y from -1 to 1`

#### Derivative and Integrals

Derivatives:
`derivative of e^(sin(x))`
`differentiate e^(sin(x))`
`(e^sin(x))'`

Antiderivatives (indefinite integrals):
`integrate sin(3x)^2`
`integral of sin(3x)^2`

Definite integrals:
`integrate sin(3x)^2 , x from 0 to pi`

#### Solving equations, and simultaneous equations

`solve x^2+3x-1=0`
`solve x^2+y^2=1 and y=2x`
`solve c1+c2=2 and 2c1+4c2=-6`
or solve the same linear equations using using matrices (see below):
`solve {{1,1},{2,4}}.{{c1},{c2}} = {{2},{-6}}`

#### Series

Infinite series:
`sum 1/n^2 from n=1 to infinity`

#### Matrix calculations

Enter a matrix:
`{{3,2,1},{-1,6,-2},{0,9,12}}`
Invert a matrix:
`{{3,2,1},{-1,6,-2},{0,9,12}}^(-1)`
Multiply two matrices:
`{{3,2},{3,-6}}.{{4,2},{-1,7}}`
[WARNING: to multiply matrices we must put a dot between them.
Otherwise Wolfram Alpha will do a "term-by-term" matrix product, which is different.]
Multiply a matrix by a column vector:
`{{1,1},{2,4}}^(-1).{{2},{-6}}`
Solve a system of simultaneous linear equations:
`solve {{1,1},{2,4}}.{{c1},{c2}} = {{2},{-6}}`

As you see from these examples, the syntax is rather flexible. Play around with the website and see what you get! Take advantage of the "extras" at the website too e.g. after integrating sin(3x)^2, Wolfram Alpha will offer to show you the steps.

Please email me with futher ideas!