Improved Euler's Method

How to adjust the code:

The function is given in the line that contains f(x,y)=1-x*4*y.
The initial condition is given by x0=0 and y0=1.
The number of iterations is given by n=80.
The step size is given by the line step=0.025.
The precision variable should be an integer. The numerical values will return with at least that many digits of precision.

Change these values to your desired values and "Evaluate".

That should be all!

2.7 #9 (a)
︠ac5f69f3-efe9-49aa-9b9c-5192dbbc0fdds︠
forget()
var('x y a b')


f(x,y)=(x+y)^(1/2)
x0=0
y0=3
n=100
step=.025
precision=5

a=numerical_approx(x0,digits=precision)
b=numerical_approx(y0,digits=precision)
print "Improved Euler's Method"
print "n", "(x(n),y(n))"
for i in range(n+1):
     print i,a,b
     aux=f(a,b)
     a=numerical_approx(a+step,digits=precision)
     b=numerical_approx(b+(aux+f(a,b+step*aux))/2*step,digits=precision)

(x, y, a, b)
Improved Euler's Method
n (x(n),y(n))
0.00000 3.0000
0.025000 3.0435
0.050000 3.0876
0.075000 3.1321
0.10000 3.1771
0.12500 3.2226
0.15000 3.2686
0.17500 3.3151
0.20000 3.3620
0.22500 3.4094
0.25000 3.4573
0.27500 3.5057
0.30000 3.5545
0.32500 3.6039
0.35000 3.6536
0.37500 3.7039
0.40000 3.7546
0.42500 3.8058
0.45000 3.8575
0.47500 3.9096
0.50000 3.9622
0.52500 4.0152
0.55000 4.0687
0.57500 4.1227
0.60000 4.1771
0.62500 4.2319
0.65000 4.2873
0.67500 4.3430
0.70000 4.3993
0.72500 4.4559
0.75000 4.5131
0.77500 4.5707
0.80000 4.6287
0.82500 4.6871
0.85000 4.7461
0.87500 4.8054
0.89999 4.8652
0.92499 4.9255
0.94999 4.9862
0.97499 5.0473
0.99999 5.1089
1.0250 5.1709
1.0500 5.2333
1.0750 5.2962
1.1000 5.3595
1.1250 5.4233
1.1500 5.4875
1.1750 5.5521
1.2000 5.6172
1.2250 5.6827
1.2500 5.7486
1.2750 5.8149
1.3000 5.8817
1.3250 5.9489
1.3500 6.0166
1.3750 6.0846
1.4000 6.1531
1.4250 6.2220
1.4500 6.2914
1.4750 6.3612
1.5000 6.4313
1.5250 6.5020
1.5500 6.5730
1.5750 6.6445
1.6000 6.7163
1.6250 6.7887
1.6500 6.8614
1.6750 6.9345
1.7000 7.0081
1.7250 7.0821
1.7500 7.1565
1.7750 7.2313
1.8000 7.3065
1.8250 7.3822
1.8500 7.4582
1.8750 7.5347
1.9000 7.6116
1.9250 7.6889
1.9500 7.7666
1.9750 7.8448
2.0000 7.9233
2.0250 8.0023
2.0500 8.0816
2.0750 8.1614
2.1000 8.2416
2.1250 8.3222
2.1500 8.4032
2.1750 8.4846
2.2000 8.5665
2.2250 8.6487
2.2500 8.7313
2.2750 8.8144
2.3000 8.8978
2.3250 8.9817
2.3500 9.0660
2.3750 9.1506
2.4000 9.2357
2.4250 9.3212
2.4500 9.4071
2.4750 9.4934
2.5000 9.5800

#9 (b)
︠2dbe308d-f967-41f7-88ab-ba4dd05b83aes︠
forget()
var('x y a b')


f(x,y)=(x+y)^(1/2)
x0=0
y0=3
n=160
step=.0125
precision=5

a=numerical_approx(x0,digits=precision)
b=numerical_approx(y0,digits=precision)
print "Improved Euler's Method"
print "n", "(x(n),y(n))"
for i in range(n+1):
     print i,a,b
     aux=f(a,b)
     a=numerical_approx(a+step,digits=precision)
     b=numerical_approx(b+(aux+f(a,b+step*aux))/2*step,digits=precision)

(x, y, a, b)
Improved Euler's Method
n (x(n),y(n))
0.00000 3.0000
0.012500 3.0217
0.025000 3.0435
0.037500 3.0655
0.050000 3.0876
0.062500 3.1098
0.075000 3.1321
0.087500 3.1546
0.10000 3.1771
0.11250 3.1998
0.12500 3.2226
0.13750 3.2456
0.15000 3.2686
0.16250 3.2918
0.17500 3.3151
0.18750 3.3385
0.20000 3.3620
0.21250 3.3857
0.22500 3.4094
0.23750 3.4333
0.25000 3.4573
0.26250 3.4815
0.27500 3.5057
0.28750 3.5301
0.30000 3.5546
0.31250 3.5792
0.32500 3.6039
0.33750 3.6287
0.35000 3.6537
0.36250 3.6787
0.37500 3.7039
0.38750 3.7292
0.40000 3.7546
0.41250 3.7802
0.42500 3.8058
0.43750 3.8316
0.45000 3.8575
0.46250 3.8835
0.47500 3.9096
0.48750 3.9358
0.50000 3.9622
0.51250 3.9887
0.52500 4.0152
0.53750 4.0419
0.55000 4.0687
0.56250 4.0957
0.57500 4.1227
0.58750 4.1498
0.60000 4.1771
0.61250 4.2045
0.62500 4.2320
0.63750 4.2596
0.64999 4.2873
0.66249 4.3151
0.67499 4.3431
0.68749 4.3711
0.69999 4.3993
0.71249 4.4276
0.72499 4.4560
0.73749 4.4845
0.74999 4.5131
0.76249 4.5418
0.77499 4.5707
0.78749 4.5996
0.79999 4.6287
0.81249 4.6579
0.82499 4.6872
0.83749 4.7166
0.84999 4.7461
0.86249 4.7757
0.87499 4.8055
0.88749 4.8353
0.89999 4.8653
0.91249 4.8953
0.92499 4.9255
0.93749 4.9558
0.94999 4.9862
0.96249 5.0167
0.97499 5.0473
0.98749 5.0781
0.99999 5.1089
1.0125 5.1399
1.0250 5.1709
1.0375 5.2021
1.0500 5.2334
1.0625 5.2648
1.0750 5.2963
1.0875 5.3279
1.1000 5.3596
1.1125 5.3914
1.1250 5.4233
1.1375 5.4554
1.1500 5.4875
1.1625 5.5198
1.1750 5.5521
1.1875 5.5846
1.2000 5.6172
1.2125 5.6499
1.2250 5.6827
1.2375 5.7156
1.2500 5.7486
1.2625 5.7817
1.2750 5.8150
1.2875 5.8483
1.3000 5.8818
1.3125 5.9153
1.3250 5.9490
1.3375 5.9827
1.3500 6.0166
1.3625 6.0506
1.3750 6.0847
1.3875 6.1189
1.4000 6.1532
1.4125 6.1876
1.4250 6.2221
1.4375 6.2567
1.4500 6.2915
1.4625 6.3263
1.4750 6.3612
1.4875 6.3963
1.5000 6.4314
1.5125 6.4667
1.5250 6.5020
1.5375 6.5375
1.5500 6.5731
1.5625 6.6088
1.5750 6.6445
1.5875 6.6804
1.6000 6.7164
1.6125 6.7525
1.6250 6.7887
1.6375 6.8251
1.6500 6.8615
1.6625 6.8980
1.6750 6.9346
1.6875 6.9713
1.7000 7.0082
1.7125 7.0451
1.7250 7.0822
1.7375 7.1193
1.7500 7.1566
1.7625 7.1939
1.7750 7.2314
1.7875 7.2689
1.8000 7.3066
1.8125 7.3444
1.8250 7.3823
1.8375 7.4202
1.8500 7.4583
1.8625 7.4965
1.8750 7.5348
1.8875 7.5732
1.9000 7.6117
1.9125 7.6503
1.9250 7.6890
1.9375 7.7278
1.9500 7.7667
1.9625 7.8057
1.9750 7.8449
1.9875 7.8841
2.0000 7.9234

Runge- Kutta Method

How to adjust the code:

The function is given in the line that contains f(x,y)=1-x*4*y.
The initial condition is given by x0=0 and y0=1.
The number of iterations is given by n=80.
The step size is given by the line step=0.025.
The precision variable should be an integer. The numerical values will return with at least that many digits of precision.

Change these values to your desired values and "Evaluate".

That should be all!

#9 (c)
︠90ca95d5-d2f5-4b5d-b0d7-3fe2865d23a8s︠
forget()
var('x y a b k')


f(x,y)=(x+y)^(1/2)
x0=0
y0=3
n=20
step=.1
precision=5


k=list(0,..,3)
a=numerical_approx(x0,digits=precision)
b=numerical_approx(y0,digits=precision)

print "Runge- Kutta Method"
print "n", "(x(n),y(n))","          ","k_(n1), k_(n2),k_(n3), k_(n4)"
for i in range(n+1):
     k[0]=f(a,b)
     k[1]=f(a+step/2,b+step/2*k[0])
     k[2]=f(a+step/2,b+step/2*k[1])
     k[3]=f(a+step,b+step*k[2])
     print i,a,b,"          ",k[0],k[1],k[2],k[3]
     a=numerical_approx(a+step,digits=precision)
     b=numerical_approx(b+step*(k[0]+2*k[1]+2*k[2]+k[3])/6,digits=precision)

(x, y, a, b, k)
Runge- Kutta Method
n (x(n),y(n))            k_(n1), k_(n2),k_(n3), k_(n4)
0.00000 3.0000            1.7320 1.7710 1.7716 1.8103
0.10000 3.1771            1.8103 1.8487 1.8492 1.8873
0.20000 3.3620            1.8873 1.9252 1.9257 1.9633
0.30000 3.5546            1.9633 2.0007 2.0011 2.0383
0.40000 3.7546            2.0383 2.0752 2.0757 2.1124
0.50000 3.9622            2.1124 2.1489 2.1493 2.1857
0.60000 4.1771            2.1857 2.2218 2.2222 2.2582
0.70000 4.3993            2.2582 2.2939 2.2943 2.3300
0.80000 4.6287            2.3300 2.3654 2.3658 2.4011
0.90000 4.8653            2.4011 2.4363 2.4366 2.4716
1.0000 5.1089            2.4716 2.5065 2.5068 2.5416
1.1000 5.3596            2.5416 2.5762 2.5765 2.6110
1.2000 5.6172            2.6110 2.6453 2.6456 2.6799
1.3000 5.8817            2.6799 2.7140 2.7143 2.7483
1.4000 6.1532            2.7483 2.7822 2.7825 2.8163
1.5000 6.4314            2.8163 2.8500 2.8502 2.8838
1.6000 6.7164            2.8838 2.9173 2.9176 2.9510
1.7000 7.0081            2.9510 2.9842 2.9845 3.0177
1.8000 7.3066            3.0177 3.0508 3.0511 3.0841
1.9000 7.6117            3.0841 3.1170 3.1173 3.1501
2.0000 7.9234            3.1501 3.1829 3.1832 3.2159

#9 (d)
︠43507292-1cdf-4e0f-86ef-127801ebb115s︠
forget()
var('x y a b k')


f(x,y)=(x+y)^(1/2)
x0=0
y0=3
n=40
step=.05
precision=5


k=list(0,..,3)
a=numerical_approx(x0,digits=precision)
b=numerical_approx(y0,digits=precision)

print "Runge- Kutta Method"
print "n", "(x(n),y(n))","          ","k_(n1), k_(n2),k_(n3), k_(n4)"
for i in range(n+1):
     k[0]=f(a,b)
     k[1]=f(a+step/2,b+step/2*k[0])
     k[2]=f(a+step/2,b+step/2*k[1])
     k[3]=f(a+step,b+step*k[2])
     print i,a,b,"          ",k[0],k[1],k[2],k[3]
     a=numerical_approx(a+step,digits=precision)
     b=numerical_approx(b+step*(k[0]+2*k[1]+2*k[2]+k[3])/6,digits=precision)

(x, y, a, b, k)
Runge- Kutta Method
n (x(n),y(n))            k_(n1), k_(n2),k_(n3), k_(n4)
0.00000 3.0000            1.7320 1.7517 1.7518 1.7713
0.050000 3.0876            1.7713 1.7908 1.7909 1.8103
0.10000 3.1771            1.8103 1.8296 1.8297 1.8489
0.15000 3.2686            1.8489 1.8681 1.8682 1.8873
0.20000 3.3620            1.8873 1.9064 1.9065 1.9254
0.25000 3.4573            1.9254 1.9443 1.9445 1.9633
0.30000 3.5546            1.9633 1.9821 1.9822 2.0009
0.35000 3.6537            2.0009 2.0196 2.0197 2.0383
0.40000 3.7546            2.0383 2.0568 2.0570 2.0755
0.45000 3.8575            2.0755 2.0939 2.0940 2.1124
0.50000 3.9622            2.1124 2.1307 2.1308 2.1491
0.55000 4.0687            2.1491 2.1674 2.1675 2.1857
0.60000 4.1771            2.1857 2.2038 2.2039 2.2220
0.65000 4.2873            2.2220 2.2401 2.2402 2.2582
0.70000 4.3993            2.2582 2.2761 2.2762 2.2941
0.75000 4.5131            2.2941 2.3120 2.3121 2.3300
0.80000 4.6287            2.3300 2.3478 2.3479 2.3656
0.85000 4.7461            2.3656 2.3833 2.3834 2.4011
0.90000 4.8653            2.4011 2.4187 2.4188 2.4364
0.95000 4.9862            2.4364 2.4540 2.4541 2.4716
1.0000 5.1089            2.4716 2.4891 2.4892 2.5067
1.0500 5.2334            2.5067 2.5241 2.5242 2.5416
1.1000 5.3596            2.5416 2.5589 2.5590 2.5763
1.1500 5.4875            2.5763 2.5936 2.5937 2.6110
1.2000 5.6172            2.6110 2.6282 2.6283 2.6455
1.2500 5.7486            2.6455 2.6627 2.6627 2.6799
1.3000 5.8817            2.6799 2.6970 2.6971 2.7141
1.3500 6.0166            2.7141 2.7312 2.7313 2.7483
1.4000 6.1532            2.7483 2.7653 2.7654 2.7823
1.4500 6.2914            2.7823 2.7993 2.7994 2.8163
1.5000 6.4314            2.8163 2.8332 2.8332 2.8501
1.5500 6.5731            2.8501 2.8669 2.8670 2.8838
1.6000 6.7164            2.8838 2.9006 2.9007 2.9174
1.6500 6.8614            2.9174 2.9342 2.9342 2.9510
1.7000 7.0081            2.9510 2.9676 2.9677 2.9844
1.7500 7.1565            2.9844 3.0010 3.0011 3.0177
1.8000 7.3066            3.0177 3.0343 3.0344 3.0509
1.8500 7.4583            3.0509 3.0675 3.0676 3.0841
1.9000 7.6117            3.0841 3.1006 3.1007 3.1172
1.9500 7.7667            3.1172 3.1336 3.1337 3.1501
2.0000 7.9234            3.1501 3.1666 3.1666 3.1830

#15 (a)

var('x','y')
plot_slope_field(x^2+y^2, (x,-5,5), (y,-5,5))

(x, y)

︠3810ceb7-9865-4833-9773-4455b0e90acfs︠
#15 (b)

forget()
var('x y a b k')


f(x,y)=x^2+y^2
x0=0
y0=1
n=80
step=.0125
precision=5


k=list(0,..,3)
a=numerical_approx(x0,digits=precision)
b=numerical_approx(y0,digits=precision)

print "Runge- Kutta Method"
print "n", "(x(n),y(n))","          ","k_(n1), k_(n2),k_(n3), k_(n4)"
for i in range(n+1):
     k[0]=f(a,b)
     k[1]=f(a+step/2,b+step/2*k[0])
     k[2]=f(a+step/2,b+step/2*k[1])
     k[3]=f(a+step,b+step*k[2])
     print i,a,b,"          ",k[0],k[1],k[2],k[3]
     a=numerical_approx(a+step,digits=precision)
     b=numerical_approx(b+step*(k[0]+2*k[1]+2*k[2]+k[3])/6,digits=precision)

(x, y, a, b, k)
Runge- Kutta Method
n (x(n),y(n))            k_(n1), k_(n2),k_(n3), k_(n4)
0.00000 1.0000            1 1.0126 1.0127 1.0256
0.012500 1.0127            1.0256 1.0389 1.0390 1.0526
0.025000 1.0256            1.0526 1.0665 1.0666 1.0809
0.037500 1.0390            1.0809 1.0955 1.0957 1.1106
0.050000 1.0527            1.1106 1.1259 1.1262 1.1419
0.062500 1.0667            1.1419 1.1580 1.1582 1.1747
0.075000 1.0812            1.1747 1.1916 1.1918 1.2091
0.087500 1.0961            1.2091 1.2269 1.2271 1.2453
0.10000 1.1115            1.2453 1.2640 1.2643 1.2834
0.11250 1.1273            1.2834 1.3030 1.3033 1.3233
0.12500 1.1436            1.3233 1.3439 1.3442 1.3653
0.13750 1.1604            1.3653 1.3870 1.3873 1.4095
0.15000 1.1777            1.4095 1.4322 1.4325 1.4559
0.16250 1.1956            1.4559 1.4798 1.4801 1.5047
0.17500 1.2141            1.5047 1.5298 1.5302 1.5560
0.18750 1.2332            1.5560 1.5825 1.5829 1.6101
0.20000 1.2530            1.6100 1.6379 1.6383 1.6669
0.21250 1.2735            1.6669 1.6963 1.6967 1.7269
0.22500 1.2947            1.7269 1.7578 1.7583 1.7901
0.23750 1.3167            1.7900 1.8226 1.8232 1.8567
0.25000 1.3395            1.8567 1.8911 1.8916 1.9270
0.26250 1.3631            1.9270 1.9633 1.9639 2.0012
0.27500 1.3877            2.0012 2.0396 2.0402 2.0797
0.28750 1.4132            2.0797 2.1202 2.1209 2.1627
0.30000 1.4397            2.1626 2.2055 2.2063 2.2505
0.31250 1.4672            2.2505 2.2959 2.2967 2.3435
0.32500 1.4960            2.3435 2.3916 2.3925 2.4422
0.33750 1.5259            2.4421 2.4932 2.4942 2.5469
0.35000 1.5570            2.5468 2.6011 2.6021 2.6581
0.36250 1.5896            2.6581 2.7157 2.7169 2.7764
0.37500 1.6235            2.7764 2.8378 2.8391 2.9024
0.38750 1.6590            2.9024 2.9678 2.9692 3.0368
0.40000 1.6961            3.0368 3.1066 3.1081 3.1802
0.41250 1.7350            3.1802 3.2548 3.2564 3.3336
0.42500 1.7757            3.3336 3.4133 3.4151 3.4978
0.43750 1.8183            3.4978 3.5832 3.5852 3.6738
0.45000 1.8631            3.6738 3.7656 3.7677 3.8629
0.46250 1.9102            3.8629 3.9616 3.9639 4.0664
0.47500 1.9598            4.0664 4.1726 4.1752 4.2857
0.48750 2.0120            4.2857 4.4003 4.4032 4.5225
0.50000 2.0670            4.5225 4.6464 4.6497 4.7788
0.51250 2.1251            4.7787 4.9130 4.9166 5.0567
0.52500 2.1866            5.0566 5.2024 5.2065 5.3587
0.53750 2.2516            5.3587 5.5174 5.5219 5.6879
0.55000 2.3206            5.6878 5.8610 5.8661 6.0474
0.56250 2.3939            6.0473 6.2368 6.2426 6.4413
0.57500 2.4720            6.4412 6.6490 6.6556 6.8739
0.58750 2.5551            6.8738 7.1026 7.1100 7.3508
0.60000 2.6440            7.3507 7.6032 7.6117 7.8780
0.61250 2.7391            7.8779 8.1577 8.1675 8.4631
0.62500 2.8412            8.4629 8.7741 8.7854 9.1148
0.63750 2.9510            9.1146 9.4620 9.4751 9.8437
0.64999 3.0694            9.8435 10.233 10.248 10.663
0.66249 3.1974            10.662 11.101 11.119 11.587
0.67499 3.3364            11.587 12.084 12.105 12.637
0.68749 3.4876            12.636 13.202 13.227 13.834
0.69999 3.6529            13.834 14.481 14.512 15.209
0.71249 3.8342            15.209 15.956 15.992 16.800
0.72499 4.0340            16.799 17.666 17.711 18.652
0.73749 4.2552            18.651 19.666 19.721 20.828
0.74999 4.5016            20.827 22.025 22.095 23.409
0.76249 4.7776            23.407 24.836 24.924 26.500
0.77499 5.0889            26.498 28.220 28.334 30.247
0.78749 5.4428            30.244 32.347 32.495 34.850
0.79999 5.8486            34.846 37.451 37.648 40.592
0.81249 6.3186            40.585 43.865 44.135 47.882
0.82499 6.8696            47.872 52.083 52.461 57.333
0.83749 7.5244            57.318 62.848 63.394 69.892
0.84999 8.3154            69.869 77.332 78.151 87.091
0.86249 9.2903            87.053 97.469 98.753 111.53
0.87499 10.522            111.47 126.63 128.76 147.95
0.88749 12.126            147.83 171.10 174.92 205.66
0.89999 14.304            205.42 243.81 251.35 305.20
0.91249 17.431            304.69 374.71 391.83 499.45
0.92499 22.301            498.17 646.75 694.81 960.99
0.93749 30.930            957.56 1363.6 1557.4 2540.9
0.94999 50.390            2540.0 4391.9 6059.9 15912.
0.96249 132.38            17525. 58523. 248150. 1.0461e7
0.97499 23240.            5.4009e8 1.1552e13 5.2127e21 4.2457e39
0.98749 8.8452e36            7.8238e73 2.3911e143 2.2333e282 7.7930e560
0.99999 1.6235e558            2.6359e1116 2.7140e2228 2.8774e4452 1.2936e8901