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.
In[1]:=
<<Calculus`DSolve`
In the first example we are looking for the general solution.
In[2]:=
DSolve[x y'[x] + y[x] == 1, y[x], x]
Out[2]=
C[1]
{{y[x] -> 1 + ----}}
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.
In[3]:=
DSolve[{y''[x] + (1/4) y'[x] + y[x] == 0, y[0] == 3, y'[0] == 2},
y[x], x]
Out[3]=
((-1 - 3 I Sqrt[7]) x)/8
(63 + 19 I Sqrt[7]) E
{{y[x] -> --------------------------------------------- +
42
((-1 + 3 I Sqrt[7]) x)/8
(63 - 19 I Sqrt[7]) E
---------------------------------------------}}
42
In[4]:=
Plot[y[x] /. %, {x,0,10}]
Out[5]=
-Graphics-
Often Mathematica can evaluate the differential equation but then not solve for y in terms of x.
In[6]:=
DSolve[{y'[x] == (2 x - 3 y[x])/(-x + y[x]), y[1] == 2},
y[x],x]
Out[6]=
x + y[x]
2 ArcTanh[---------] 2 2
Sqrt[3] x Log[-2 x + 2 x y[x] + y[x] ]
{Solve[-------------------- + ----------------------------- ==
Sqrt[3] 2
4 Sqrt[3] ArcTanh[Sqrt[3]] + 3 Log[6]
-------------------------------------, y[x]]}
6
If DSolve fails, often a numerical approximation of the solution can be obtained.
In[7]:=
NDSolve[{y'[x] == Cos[y[x]] + Sin[x]/(1+x), y[0] == 1},
y[x], {x,0,20}]
Out[7]=
{{y[x] -> InterpolatingFunction[{0., 20.}, <>][x]}}
In[8]:=
Plot[Evaluate[y[x] /. %] , {x,0,20}]
Out[9]=
-Graphics-
It is also possible to plot direction fields. Suppose the differential equation is y'[x] == Cos[x] y + Sin[x] y^2.
In[10]:=
<<Graphics`PlotField`;
PlotVectorField[{1, Cos[x] y + Sin[x] y^2}, {x,0,6},
{y,-2,2}, ScaleFunction -> (.5 &)]
Out[11]=
-Graphics-
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