Mathematica can solve many of the ODE's encountered in your class. The main package is called DSolve. This routine is built into Mathematica but a more powerful version exist which can be loaded.
<<Calculus`DSolve`
In the first example we are looking for the general solution.
DSolve[x y'[x] + y[x] == 1, y[x], x]
In the second example we will include initial conditions. Notice that the initial conditions are included with the equation. The answer looks strange, but a plot of the solution gives an idea of the solution.
DSolve[{y''[x] + (1/4) y'[x] + y[x] == 0, y[0] == 3, y'[0] == 2},
y[x], x]
Plot[y[x] /. %, {x,0,10}]Often Mathematica can evaluate the differential equation but then not solve for y in terms of x.
DSolve[{y'[x] == (2 x - 3 y[x])/(-x + y[x]), y[1] == 2},
y[x],x]If DSolve fails, often a numerical approximation of the solution can be obtained.
NDSolve[{y'[x] == Cos[y[x]] + Sin[x]/(1+x), y[0] == 1},
y[x], {x,0,20}]
Plot[Evaluate[y[x] /. %] , {x,0,20}]It is also possible to plot direction fields. Suppose the differential equation is y'[x] == Cos[x] y + Sin[x] y^2.
<<Graphics`PlotField`;
PlotVectorField[{1, Cos[x] y + Sin[x] y^2}, {x,0,6},
{y,-2,2}, ScaleFunction -> (.5 &)]Go up to the Applied Math Page.