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differential equations - Mathematica Improved Euler's Method


Population growth in Mathematica with NDSolve:


f[r_, a_, T_] := NDSolve[{x' t] == r*x[t]*(1 - x[t]), x[0] == a}, x, {t, 0, T}]

s1 = f[0.1, 0.5, 30];

s2 = f[0.1, 2, 30];

Plot[{x[t] /. s1, x[t] /. s2, 1}, {t, 0, 30}, PlotRange -> {0, 2}]
Plot[Evaluate[Table[x[t] /. f[0.3, 0.2 n, 10], {n, 1, 10}]], PlotRange -> {0, 2}]

How can we solve it with Improved Euler's method?


P[q_, h_, N_] := (
  u[0] = 1; 
  Do[u[n + 1] = u[n] + h*f[n*h + (h*q/2), u[n] + (h*q/2)*f[n*h, u[n]]], {n, 0, N}]
 )


f[x_, t_] := r*x[t]*(1 - x[t])


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