Решение линейного дифференциального уравнения второго порядка

Найдём корни характеристического уравнения:
p2+5p+30=0
D=25-4∙16=-39
p1,2=-5±95i2
Для определения времени наблюдения T=3÷4τ, где τ=1/pmin
pmin=minp1,p2=30
h=∆t=T20=3∙1/3020≈0.03
Перепишем систему в виде:
y'=zz'=5-5z-30y;y0=5, z0=10
Число интервалов разбиения
n=20, ti=i∙h=0.03i, i=0,20
Решаем методом Рунге-Кутта четвёртого порядка:
yi+1=yi+∆yi, zi+1=zi+∆zi
∆yi=16K1i+2K2i+2K3i+K4i, ∆zi=16M1i+2M2i+2M3i+M4i
K1i=hf1ti,zi,yi, M1i=hf2ti,zi,yi,
K2i=hf1ti+h2,zi+M1i2,yi+K1i2, M2i=hf2ti+h2,zi+M1i2,yi+K1i2
K3i=hf1ti+h2,zi+M2i2,yi+K2i2, M3i=hf2ti+h2,zi+M2i2,yi+K2i2
K4i=hf1ti+h,zi+M3i,yi+K3i, M4i=hf2ti+h,zi+M3i,yi+K3i
Получаем:
ti
K1i
K2i
K3i
K4i
M1i
M2i
M3i
M4i
yi
zi
0 0.3 0.2123 0.2168 0.1341 -5.85 -5.5462 -5.5295 -5.2157 5 10
0.03 0.1339 0.0557 0.0607 -0.0123 -5.2134 -4.8827 -4.8723 -4.5372 5.2154 4.4638
0.06 -0.0124 -0.0804 -0.0752 -0.1379 -4.5351 -4.1894 -4.1847 -3.8396 5.2744 -0.413
0.09 -0.138 -0.1956 -0.1903 -0.2426 -3.8377 -3.4878 -3.4882 -3.1432 5.1975 -4.6001
0.12 -0.2427 -0.2898 -0.2846 -0.3267 -3.1416 -2.7967 -2.8014 -2.4652 5.0054 -8.0889
0.15 -0.3267 -0.3636 -0.3587 -0.3909 -2.4638 -2.132 -2.1402 -1.8199 4.7191 -10.8894
0.18 -0.3908 -0.4181 -0.4134 -0.4364 -1.8187 -1.5065 -1.5176 -1.219 4.3587 -13.0274
0.21 -0.4363 -0.4545 -0.4502 -0.4646 -1.218 -0.9304 -0.9437 -0.6713 3.9437 -14.5417
0.24 -0.4644 -0.4745 -0.4706 -0.4772 -0.6706 -0.4113 -0.4262 -0.1831 3.492 -15.4813
0.27 -0.4771 -0.4798 -0.4764 -0.4762 -0.1826 0.0458 0.0299 0.2417 3.02 -15.9028
0.3 -0.476 -0.4724 -0.4695 -0.4634 0.242 0.4381 0.4217 0.6013 2.5424 -15.8677
0.33 -0.4632 -0.4542 -0.4517 -0.4408 0.6014 0.7648 0.7484 0.8957 2.0718 -15.4405
0.36 -0.4406 -0.4272 -0.4252 -0.4103 0.8957 1.0268 1.0109 1.1267 1.6192 -14.6866
0.39 -0.4101 -0.3932 -0.3917 -0.3738 1.1266 1.2266 1.2115 1.2974 1.1933 -13.6703
0.42 -0.3736 -0.3541 -0.3531 -0.333 1.2971 1.368 1.3539 1.4118 0.801 -12.4535
0.45 -0.3328 -0.3117 -0.311 -0.2896 1.4115 1.4554 1.4426 1.475 0.4475 -11.0947
0.48 -0.2894 -0.2673 -0.267 -0.2449 1.4746 1.4942 1.4828 1.4925 0.1362 -9.6477
0.51 -0.2448 -0.2224 -0.2225 -0.2004 1.492 1.4903 1.4803 1.4702 -0.131 -8.1608
0.54 -0.2003 -0.1783 -0.1786 -0.1571 1.4697 1.4496 1.4412 1.4142 -0.3535 -6.6769
0.57 -0.157 -0.1358 -0.1363 -0.1158 1.4137 1.3783 1.3714 1.3307 -0.532 -5.2327
0.6
-0.6682 -3.8587
Строим график:
Для вычислений использовали программу в MathCad:
K1i
K2i
K3i
K4i
M1i
M2i
M3i
M4i