import math
def runge_kutta_4(f, x0, y0, x_final, step_size):
"""
Implements the Runge-Kutta 4th order method for solving ODEs.
"""
x_vals = [x0 + i * step_size for i in range(int((x_final - x0) / step_size) + 1)]
y = y0
print("\nRunge-Kutta 4 Method")
print("x\t\t y\t\t k1\t\t k2\t\t k3\t\t k4")
for x in x_vals:
k1 = f(x, y)
k2 = f(x + step_size / 2, y + step_size * k1 / 2)
k3 = f(x + step_size / 2, y + step_size * k2 / 2)
k4 = f(x + step_size, y + step_size * k3)
print(f"{x:.2f}\t {y:.4f}\t {k1:.4f}\t {k2:.4f}\t {k3:.4f}\t {k4:.4f}")
y += (step_size / 6) * (k1 + 2 * k2 + 2 * k3 + k4)
# Define the differential equations as functions
def eq1(x, y): return 2*x - 3*y + 1
def eq2(x, y): return 4*x - 2*y
def eq3(x, y): return 1 - y**2
def eq4(x, y): return x**2 - y**2
def eq5(x, y): return math.exp(-y)
def eq6(x, y): return x - y**2
def eq7(x, y): return (x - y)**2
def eq8(x, y): return x*y + math.sqrt(y) if y >= 0 else x*y
def eq9(x, y): return x*y**2 - y
def eq10(x, y): return y - y**2
# Initial conditions and intervals for each problem
problems = [
(eq1, 1, 5, 1.5),
(eq2, 0, 2, 0.5),
(eq3, 0, 0, 0.5),
(eq4, 0, 1, 0.5),
(eq5, 0, 0, 0.5),
(eq6, 0, 0, 0.5),
(eq7, 0, 0.5, 0.5),
(eq8, 0, 1, 0.5),
(eq9, 1, 1, 1.5),
(eq10, 0, 0.5, 0.5)
]
# Solve for step sizes h = 0.1 and h = 0.05
for i, (func, x0, y0, x_final) in enumerate(problems, start=1):
print(f"\nSolving Problem 9.1.{i} with step size h = 0.1")
runge_kutta_4(func, x0, y0, x_final, 0.1)
print(f"\nSolving Problem 9.1.{i} with step size h = 0.05")
runge_kutta_4(func, x0, y0, x_final, 0.05)Solving Problem 9.1.1 with step size h = 0.1 Runge-Kutta 4 Method x y k1 k2 k3 k4 1.00 5.0000 -12.0000 -10.1000 -10.3850 -8.6845 1.10 3.9724 -8.7173 -7.3097 -7.5208 -6.2610 1.20 3.2284 -6.2853 -5.2425 -5.3989 -4.4656 1.30 2.6945 -4.4836 -3.7111 -3.8270 -3.1355 1.40 2.3163 -3.1489 -2.5765 -2.6624 -2.1501 1.50 2.0533 -2.1600 -1.7360 -1.7996 -1.4201 Solving Problem 9.1.1 with step size h = 0.05 Runge-Kutta 4 Method x y k1 k2 k3 k4 1.00 5.0000 -12.0000 -11.0500 -11.1212 -10.2318 1.05 4.4452 -10.2356 -9.4180 -9.4793 -8.7137 1.10 3.9723 -8.7170 -8.0133 -8.0660 -7.4071 1.15 3.5700 -7.4100 -6.8042 -6.8497 -6.2825 1.20 3.2283 -6.2850 -5.7636 -5.8027 -5.3146 1.25 2.9389 -5.3167 -4.8679 -4.9016 -4.4814 1.30 2.6944 -4.4832 -4.0970 -4.1260 -3.7643 1.35 2.4886 -3.7659 -3.4335 -3.4584 -3.1471 1.40 2.3162 -3.1485 -2.8623 -2.8838 -2.6159 1.45 2.1724 -2.6171 -2.3708 -2.3893 -2.1587 1.50 2.0532 -2.1597 -1.9477 -1.9636 -1.7651 Solving Problem 9.1.2 with step size h = 0.1 Runge-Kutta 4 Method x y k1 k2 k3 k4 0.00 2.0000 -4.0000 -3.4000 -3.4600 -2.9080 0.10 1.6562 -2.9124 -2.4212 -2.4703 -2.0183 0.20 1.4110 -2.0219 -1.6198 -1.6600 -1.2900 0.30 1.2465 -1.2929 -0.9636 -0.9965 -0.6936 0.40 1.1480 -0.6960 -0.4264 -0.4534 -0.2053 0.50 1.1037 -0.2073 0.0134 -0.0087 0.1944 Solving Problem 9.1.2 with step size h = 0.05 Runge-Kutta 4 Method x y k1 k2 k3 k4 0.00 2.0000 -4.0000 -3.7000 -3.7150 -3.4285 0.05 1.8145 -3.4290 -3.1576 -3.1711 -2.9119 0.10 1.6562 -2.9124 -2.6668 -2.6790 -2.4445 0.15 1.5225 -2.4449 -2.2227 -2.2338 -2.0215 0.20 1.4110 -2.0219 -1.8208 -1.8309 -1.6388 0.25 1.3196 -1.6392 -1.4572 -1.4663 -1.2926 0.30 1.2464 -1.2929 -1.1282 -1.1365 -0.9792 0.35 1.1898 -0.9795 -0.8305 -0.8380 -0.6957 0.40 1.1480 -0.6960 -0.5612 -0.5679 -0.4392 0.45 1.1197 -0.4394 -0.3174 -0.3235 -0.2071 0.50 1.1036 -0.2073 -0.0969 -0.1024 0.0030 Solving Problem 9.1.3 with step size h = 0.1 Runge-Kutta 4 Method x y k1 k2 k3 k4 0.00 0.0000 1.0000 0.9975 0.9975 0.9900 0.10 0.0997 0.9901 0.9777 0.9779 0.9610 0.20 0.1974 0.9610 0.9398 0.9403 0.9151 0.30 0.2913 0.9151 0.8864 0.8874 0.8556 0.40 0.3799 0.8556 0.8213 0.8227 0.7863 0.50 0.4621 0.7864 0.7486 0.7505 0.7115 Solving Problem 9.1.3 with step size h = 0.05 Runge-Kutta 4 Method x y k1 k2 k3 k4 0.00 0.0000 1.0000 0.9994 0.9994 0.9975 0.05 0.0500 0.9975 0.9944 0.9944 0.9901 0.10 0.0997 0.9901 0.9845 0.9846 0.9778 0.15 0.1489 0.9778 0.9700 0.9700 0.9610 0.20 0.1974 0.9610 0.9510 0.9511 0.9400 0.25 0.2449 0.9400 0.9280 0.9281 0.9151 0.30 0.2913 0.9151 0.9013 0.9015 0.8868 0.35 0.3364 0.8869 0.8714 0.8717 0.8556 0.40 0.3799 0.8556 0.8389 0.8393 0.8220 0.45 0.4219 0.8220 0.8042 0.8046 0.7864 0.50 0.4621 0.7864 0.7679 0.7683 0.7495 Solving Problem 9.1.4 with step size h = 0.1 Runge-Kutta 4 Method x y k1 k2 k3 k4 0.00 1.0000 -1.0000 -0.9000 -0.9095 -0.8164 0.10 0.9094 -0.8170 -0.7319 -0.7393 -0.6580 0.20 0.8358 -0.6585 -0.5821 -0.5882 -0.5137 0.30 0.7772 -0.5141 -0.4423 -0.4477 -0.3765 0.40 0.7327 -0.3769 -0.3071 -0.3121 -0.2421 0.50 0.7018 -0.2425 -0.1731 -0.1779 -0.1078 Solving Problem 9.1.4 with step size h = 0.05 Runge-Kutta 4 Method x y k1 k2 k3 k4 0.00 1.0000 -1.0000 -0.9500 -0.9524 -0.9045 0.05 0.9524 -0.9046 -0.8589 -0.8610 -0.8170 0.10 0.9094 -0.8170 -0.7747 -0.7766 -0.7354 0.15 0.8706 -0.7355 -0.6957 -0.6974 -0.6585 0.20 0.8358 -0.6585 -0.6207 -0.6222 -0.5850 0.25 0.8047 -0.5851 -0.5486 -0.5500 -0.5140 0.30 0.7772 -0.5141 -0.4787 -0.4800 -0.4449 0.35 0.7533 -0.4449 -0.4102 -0.4114 -0.3768 0.40 0.7327 -0.3769 -0.3425 -0.3438 -0.3095 0.45 0.7156 -0.3095 -0.2754 -0.2766 -0.2424 0.50 0.7018 -0.2425 -0.2084 -0.2096 -0.1754 Solving Problem 9.1.5 with step size h = 0.1 Runge-Kutta 4 Method x y k1 k2 k3 k4 0.00 0.0000 1.0000 0.9512 0.9536 0.9090 0.10 0.0953 0.9091 0.8687 0.8705 0.8333 0.20 0.1823 0.8333 0.7993 0.8007 0.7692 0.30 0.2624 0.7692 0.7402 0.7413 0.7143 0.40 0.3365 0.7143 0.6892 0.6901 0.6667 0.50 0.4055 0.6667 0.6448 0.6455 0.6250 Solving Problem 9.1.5 with step size h = 0.05 Runge-Kutta 4 Method x y k1 k2 k3 k4 0.00 0.0000 1.0000 0.9753 0.9759 0.9524 0.05 0.0488 0.9524 0.9300 0.9305 0.9091 0.10 0.0953 0.9091 0.8887 0.8891 0.8696 0.15 0.1398 0.8696 0.8509 0.8513 0.8333 0.20 0.1823 0.8333 0.8162 0.8165 0.8000 0.25 0.2231 0.8000 0.7842 0.7845 0.7692 0.30 0.2624 0.7692 0.7546 0.7549 0.7407 0.35 0.3001 0.7407 0.7271 0.7274 0.7143 0.40 0.3365 0.7143 0.7016 0.7019 0.6897 0.45 0.3716 0.6897 0.6779 0.6781 0.6667 0.50 0.4055 0.6667 0.6556 0.6558 0.6452 Solving Problem 9.1.6 with step size h = 0.1 Runge-Kutta 4 Method x y k1 k2 k3 k4 0.00 0.0000 0.0000 0.0500 0.0500 0.1000 0.10 0.0050 0.1000 0.1499 0.1498 0.1996 0.20 0.0200 0.1996 0.2491 0.2489 0.2980 0.30 0.0449 0.2980 0.3464 0.3461 0.3937 0.40 0.0795 0.3937 0.4402 0.4397 0.4848 0.50 0.1235 0.4848 0.5282 0.5275 0.5689 Solving Problem 9.1.6 with step size h = 0.05 Runge-Kutta 4 Method x y k1 k2 k3 k4 0.00 0.0000 0.0000 0.0250 0.0250 0.0500 0.05 0.0012 0.0500 0.0750 0.0750 0.1000 0.10 0.0050 0.1000 0.1249 0.1249 0.1499 0.15 0.0112 0.1499 0.1748 0.1748 0.1996 0.20 0.0200 0.1996 0.2244 0.2243 0.2490 0.25 0.0312 0.2490 0.2736 0.2736 0.2980 0.30 0.0449 0.2980 0.3223 0.3222 0.3463 0.35 0.0610 0.3463 0.3701 0.3701 0.3937 0.40 0.0795 0.3937 0.4170 0.4169 0.4399 0.45 0.1003 0.4399 0.4626 0.4625 0.4848 0.50 0.1235 0.4848 0.5066 0.5065 0.5279 Solving Problem 9.1.7 with step size h = 0.1 Runge-Kutta 4 Method x y k1 k2 k3 k4 0.00 0.5000 0.2500 0.2139 0.2122 0.1774 0.10 0.5213 0.1775 0.1446 0.1433 0.1127 0.20 0.5358 0.1127 0.0849 0.0841 0.0596 0.30 0.5443 0.0597 0.0389 0.0385 0.0219 0.40 0.5482 0.0220 0.0099 0.0097 0.0024 0.50 0.5493 0.0024 0.0000 0.0000 0.0026 Solving Problem 9.1.7 with step size h = 0.05 Runge-Kutta 4 Method x y k1 k2 k3 k4 0.00 0.5000 0.2500 0.2316 0.2312 0.2130 0.05 0.5116 0.2130 0.1953 0.1949 0.1775 0.10 0.5213 0.1775 0.1606 0.1603 0.1439 0.15 0.5294 0.1439 0.1281 0.1278 0.1127 0.20 0.5358 0.1127 0.0983 0.0981 0.0845 0.25 0.5407 0.0845 0.0717 0.0715 0.0597 0.30 0.5443 0.0597 0.0487 0.0486 0.0387 0.35 0.5467 0.0387 0.0298 0.0297 0.0220 0.40 0.5482 0.0220 0.0153 0.0153 0.0098 0.45 0.5490 0.0098 0.0055 0.0055 0.0024 0.50 0.5493 0.0024 0.0006 0.0006 0.0000 Solving Problem 9.1.8 with step size h = 0.1 Runge-Kutta 4 Method x y k1 k2 k3 k4 0.00 1.0000 1.0000 1.0772 1.0793 1.1634 0.10 1.1079 1.1634 1.2548 1.2576 1.3575 0.20 1.2337 1.3575 1.4663 1.4700 1.5892 0.30 1.3807 1.5892 1.7194 1.7244 1.8675 0.40 1.5531 1.8675 2.0241 2.0306 2.2033 0.50 1.7561 2.2032 2.3925 2.4012 2.6106 Solving Problem 9.1.8 with step size h = 0.05 Runge-Kutta 4 Method x y k1 k2 k3 k4 0.00 1.0000 1.0000 1.0380 1.0385 1.0782 0.05 1.0519 1.0782 1.1196 1.1202 1.1634 0.10 1.1079 1.1634 1.2084 1.2091 1.2562 0.15 1.1684 1.2562 1.3053 1.3061 1.3575 0.20 1.2337 1.3575 1.4111 1.4120 1.4681 0.25 1.3043 1.4681 1.5268 1.5278 1.5892 0.30 1.3807 1.5892 1.6534 1.6546 1.7219 0.35 1.4634 1.7219 1.7923 1.7937 1.8675 0.40 1.5531 1.8675 1.9447 1.9463 2.0274 0.45 1.6504 2.0274 2.1123 2.1141 2.2032 0.50 1.7561 2.2032 2.2967 2.2988 2.3969 Solving Problem 9.1.9 with step size h = 0.1 Runge-Kutta 4 Method x y k1 k2 k3 k4 1.00 1.0000 0.0000 0.0500 0.0528 0.1064 1.10 1.0052 0.1063 0.1638 0.1676 0.2313 1.20 1.0219 0.2312 0.3015 0.3071 0.3877 1.30 1.0525 0.3875 0.4791 0.4878 0.5966 1.40 1.1011 0.5963 0.7236 0.7382 0.8958 1.50 1.1747 0.8952 1.0855 1.1121 1.3598 Solving Problem 9.1.9 with step size h = 0.05 Runge-Kutta 4 Method x y k1 k2 k3 k4 1.00 1.0000 0.0000 0.0250 0.0257 0.0514 1.05 1.0013 0.0514 0.0779 0.0787 0.1063 1.10 1.0052 0.1063 0.1349 0.1358 0.1657 1.15 1.0120 0.1657 0.1971 0.1982 0.2312 1.20 1.0219 0.2312 0.2660 0.2674 0.3044 1.25 1.0352 0.3044 0.3437 0.3454 0.3876 1.30 1.0525 0.3875 0.4327 0.4347 0.4836 1.35 1.0742 0.4836 0.5362 0.5388 0.5964 1.40 1.1011 0.5963 0.6588 0.6622 0.7311 1.45 1.1342 0.7311 0.8066 0.8111 0.8953 1.50 1.1747 0.8952 0.9883 0.9944 1.0994 Solving Problem 9.1.10 with step size h = 0.1 Runge-Kutta 4 Method x y k1 k2 k3 k4 0.00 0.5000 0.2500 0.2498 0.2498 0.2494 0.10 0.5250 0.2494 0.2486 0.2486 0.2475 0.20 0.5498 0.2475 0.2461 0.2461 0.2445 0.30 0.5744 0.2445 0.2425 0.2425 0.2403 0.40 0.5987 0.2403 0.2377 0.2378 0.2350 0.50 0.6225 0.2350 0.2320 0.2320 0.2288 Solving Problem 9.1.10 with step size h = 0.05 Runge-Kutta 4 Method x y k1 k2 k3 k4 0.00 0.5000 0.2500 0.2500 0.2500 0.2498 0.05 0.5125 0.2498 0.2496 0.2496 0.2494 0.10 0.5250 0.2494 0.2490 0.2490 0.2486 0.15 0.5374 0.2486 0.2481 0.2481 0.2475 0.20 0.5498 0.2475 0.2469 0.2469 0.2461 0.25 0.5622 0.2461 0.2453 0.2453 0.2445 0.30 0.5744 0.2445 0.2435 0.2435 0.2425 0.35 0.5866 0.2425 0.2414 0.2414 0.2403 0.40 0.5987 0.2403 0.2390 0.2390 0.2378 0.45 0.6106 0.2378 0.2364 0.2364 0.2350 0.50 0.6225 0.2350 0.2335 0.2335 0.2320