/* Here is how to solve a differential equation */
'diff(y, x) = x * sqrt(y);
ode2(%, y, x);
ic1(%, x = 0, y = 1);
factor(solve(%, y)); /* [y = (x^2 + 4)^2 / 16] */

/* The Runge-Kutta solver is builtin */

load(dynamics)$
sol: rk(t * sqrt(y), y, 1, [t, 0, 10, 1.0])$
plot2d([discrete, sol])$

/* An implementation of RK4 for one equation */

rk4(f, x0, y0, x1, n) := block([h, x, y, vx, vy, k1, k2, k3, k4],
   h: bfloat((x1 - x0) / (n - 1)),
   x: x0,
   y: y0,
   vx: makelist(0, n + 1),
   vy: makelist(0, n + 1),
   vx[1]: x0,
   vy[1]: y0,
   for i from 1 thru n do (
      k1: bfloat(h * f(x, y)),
      k2: bfloat(h * f(x + h / 2, y + k1 / 2)),
      k3: bfloat(h * f(x + h / 2, y + k2 / 2)),
      k4: bfloat(h * f(x + h, y + k3)),
      vy[i + 1]: y: y + (k1 + 2 * k2 + 2 * k3 + k4) / 6,
      vx[i + 1]: x: x + h
   ),
   [vx, vy]
)$

[x, y]: rk4(lambda([x, y], x * sqrt(y)), 0, 1, 10, 101)$

plot2d([discrete, x, y])$

s: map(lambda([x], (x^2 + 4)^2 / 16), x)$

for i from 1 step 10 thru 101 do print(x[i], " ", y[i], " ", y[i] - s[i]);