Topic 14.3: 4th-Order Runge Kutta's Method (Maple)

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Given the IVP y⁽¹⁾(t = t y(t) - t² + 1 with y(0) = 3, to approximate y(1), we can do:

t[0] := 0;
t[1] := 1;
y[0] := 3;
f := (t, y) -> t*y - t^2 + 1;
h := t[1] - t[0];
K[0] := f(t[0],       y[0]           );
K[1] := f(t[0] + h/2, y[0] + h/2*K[0]);
K[2] := f(t[0] + h/2, y[0] + h/2*K[1]);
K[3] := f(t[0] + h  , y[0] +   h*K[2]);
y[1] := y[0] + h*(K[0] + 2*K[1] + 2*K[2] + K[3])/6;