Needs["Graphics`PlotField`"]

\(\(\(EulerMethod[\ f_, {x0_, \ y0_}, \ h_, \ 
        n_\ ]\)\(\ \)\(:=\)\(\ \)\(
\)\(
\)\(		\)\(Module[
		\ \ \ {EMStep}, 
		\ \ \ Interpolation[
\ \ \ 	\ \ \ \ \ \ \ \ \ EMStep[{x_, 
                y_}]\  := \ 
            N[{x + h, y + h*f[x, y]}]; 
		\ \ \ \ \ \ \ \ \ \ NestList[
            EMStep, {x0, y0}, 
            n], 
			\ \ \ \ \ \ \ InterpolationOrder\  -> \ 
            1
		\ \ \ ]
	\ ]\)\(\ \ \ \ \ \ \ \ \ \ \ \ \ \)\)\)

\(RKMethod[f_, {x0_, y0_}, h_, 
        n_]\  := 
	
\ \ \ \ \ Module[
		\ \ \ \ \ \ \ {RKStep}, 
		
		\ \ \ \ \ \ \ \ RKStep[{x_, y_}]\  := \ 
			\ \ \ \ \ \ Module[{k1, 
              k2, k3, k4}, 
		\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ k1\  = \ f[x, 
                y]; 
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ k2\  = \ f[x + h/2, \ 
                y + \ k1*
                    h/2]; 
		\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ k3\  = \ f[x + h/2, y + \ k2*h/2]; 
										\ \ k4\ \  = \ 
              f[x + h, 
                y + k3*h]; 
		\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ N[{x + h, y + h*\((k1\  + \ 2*k2\  + \ 2*k3\  + \ k4)\)/6}, 
              12]
			\ \ \ \ \ ]; 
		
		\ \ \ \ \ \ \ \ Interpolation[
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ NestList[
            RKStep, {x0, y0}, 
            n], 
			\ \ \ \ \ \ \ \ \ \ \ \ InterpolationOrder\  -> \ 
            1
			\ \ \ \ \ ]
	\ \ \ \ \ ]\)

f[t_, y_]  =  (1 + y^2 - t)/(2  y)

f[t, Sqrt[t]]

esol =  EulerMethod[f, {1, 1},  .01, 2000]

rksol1  =  RKMethod[f, {1, 1},  .1, 200]

rksol2  =  RKMethod[f, {1, 1},  .01, 2000]

ndsol[t_]  =  
      y[t] /. First[
          NDSolve[{y'[t] == (1 + y[t]^2 - t)/(2  y[t]), 
              y[1] == 1}, y[t], {t, 1, 20}, 
            MaxSteps -> 5000]]

?NDSolve

Plot[{rksol1[t], rksol2[t], esol[t], ndsol[t]}, {t, 1, 20}, 
      PlotRange -> {0, 6}]

Simplify[
      DSolve[{y'[t] == (1 + y[t]^2 - t)/(2  y[t]), 
          y[1] == 1}, y[t], t]]

soln[t_, a_, b_]  :=  {x[t], y[t]} /. 
        First[
NDSolve[{x'[t]  ==  
                x[t] Cos[t]  -  y[t], y'[t] == 
                y[t] Sin[t]  +  x[t], x[0] == a, 
              y[0] == b}, 
{x[t], y[t]}, {t, -10, 
              10}]]

ParametricPlot[
      Evaluate[{soln[t, 1, 1], soln[t, -1, 0]}], {t, -10, 10}, 
      AxesLabel -> {(Subscript[x, 1]), (Subscript[x, 2])}]

Options[ParametricPlot]

A  =  {{1, -1}, {0, 1}} . {{1, 0}, {0, -2}} . 
        Inverse[{{1, -1}, {0, 1}}]

A . {x, y}

\(PlotVectorField[A . {x, y}, {x, \(-3\), 3}, {y, \(-3\), 3}, 
      Axes -> \ True, 
	ScaleFunction -> \((1 &)\), 
      Ticks -> None, Frame -> True, AspectRatio -> 1, \ 
      PlotPoints\  -> \ 15]\)