Basic help with mathematica 5.2, please?

Question

I have just bought student edition of Mathematica and I have no idea how to so some basic stuff. Could anybody please tell me how to "sketch several typical solutions of the given differential equation and highlight the one that satisfies the initial condition" using Mathematica? I have tried looking at Wolfram website but couldn't find clear explanation.
Could you please give me some example of the code for any differential equation and any initial condition you want.
Thanks!

Answer 1

The tool "DSolve" can solve differential equations. Here is an example of how to use it. (This example comes from Mathematica help)
DSolve[y'[x] == a y[x], y[x], x]

The differential equation in this example is y'[x] == a y[x]
y[x] is the function that you are solving the equation for.
x is the independent variable.

The answer that you get is:
{{y[x] -> E^(a x) C[1]}}

That is a set of solutions to the equation. In this example, the set has only one member.

The solution involves a constant C[1]. It is a constant of integration. You could assign any arbitrary value to the constant C[1], and still have a solution to the equation. It is possible to get solutions with more than one constant of integration. In that case, the constants are named C[1], C[2], C[3], ... etc.

Note: My copy of Mathematica is version 3.0. However, I believe that all of the above holds for version 5.2 also.