Demonstration 2: families of solutions curves to ordinary differential equations.

The linear first order ODE 
y'=-(2/t) + 4 t
has solutions of the form 
y  = (t^2) + (c/(t^2.))
I have plotted the graphs of these solutions for various values of c below:

f[c_, x_] := (x^2) + (c/(x^2))

\(plottable\  = \ 
      Table[
		
        Plot[f[2*i, \ x], \ {x, \ \(-2\), 2}, 
			
          PlotRange -> \ {\(-20\), 20}, 
			\ 
          DisplayFunction -> 
				Identity], \ {i, \(-10\), \ 10}]; 

    Show[plottable, \ DisplayFunction\  -> \ $DisplayFunction]\)

The differential equation
y' = 2 t y + 1
has solutions
y[t] = 
        exp [-(t^2)] ((∫_0^t) exp[(s^2)] ds) + 
          c exp[-(t^2)].
The indefinite integral cannot be expressed in terms of elementary functions,  but it can be computed numerically.  I have computed the solution curves for a number of initial values, 
y(0) =  i/2,   -10 ≤ i ≤ 10.
Since the solutions have to be computed numerically in any case, I actually used the Mathematica numericaL ODE solver  NDSolve[  ]  to obtain the solution curves.

soln[i_] := NDSolve[{y'[t]==(2 t y[t] + 1),y[0] == i},y[t],{t, -2, 2}];
\(solnlist\  = \ Table[\ soln[i/2]	, \ {i, \(-10\), 10}]\)	;
\(plotlist\  = \ 
        Map[\ 
	
          Plot[Evaluate[y[t] /. \ \ #], 
				{t, \ \(-2\), 2}, \ 
              PlotRange\  -> {\(-5\), 5}, 
			
              DisplayFunction\  -> \ Identity]	&, \ solnlist]\);

\(\(
Show[plotlist, \ DisplayFunction\  -> \ $DisplayFunction]\)\)