CCF20090601014

y₂₀ = yo + 0,5/7 P20 = 0.3 + 0.5-0,5-1,2884 = 0,6221

Pio = 5 q^x0+0'^5H sm(x₀ + 0,5/7) - 2>-2o = 5 e⁰⁺⁰'²⁵ sin(0 + 0.25) - 2-0,6221 = 0.3442

^30 ⁼ yo ⁺ h P30 ⁼ 0,3 + 0,5-0,3442 = 0,4721

P40 = 5 e^Al siavi - 2y3o = 5 e⁰” sin0.5 - 2-0.4721 = 3.0080

y, =yo + ’/₆ h (/>,o + 2/720 + 2/730 + Ało) = 0,3 + 7₆-0,5(-0,6 + 2-1,2884 + 2-03442 + 3,0080) = 0,7728

P\ 1 = 5 e^vl sinxi - 2y, = 5 e⁰'⁵ sin0.5 - 2-0,7728 = 2,4067 yu =yi + 0.5/7 p_u =0.7728 + 0,5-0.5-2,4067= 1,3744

p₂\ = 5 e^vl+0'^5/’ sinfo + 0,5h) -2y„ = 5 e^a5+a25 sin(0.5 + 0.25) - 2-1,3744= 4,4663 y₂i =y\ + 0,5/7/72, =0.7728 + 0,5-0,5-4.4663 = 1.8893

p₃\ = 5 e^xl+0'^5h sin(x, + 0,5h) - 2y_2] = 5 e°'⁵⁺⁰’²⁵ sin(0.5 + 0.25) - 2-1.8893 = 3,4365

Bi =>4 + hp₃\ =0.7728 + 0,5-3.4365 = 2.4910

p₄\ = 5 e^v2 sia\'2 - 2^31 = 5 e¹ sini - 2-2.4910 = 6.4548

y₂ =yi + '/₆ I>(P\\+ 2/72i + 2/7₃, +/741) = 0,7728 + '/₆-0,5(2,4067 + 2-4,4663 + 2-3,4365 + 6,4548) = = 2,8283

pn = 5 e^r2 siav₂ - 2y₂ = 5 e¹ sini - 2-2,8283 = 5.7801 y_i2 = y₂ + 0.5/7 p_]2 = 2,8283 + 0.5-0,5-5.7801 = 4,2734

P22 = 5 q^x2+0'^5H sin(x₂ + 0.5/7) - 2yn = 5 e^l+0'²⁵ sin(l + 0.25) - 2-4.2734 = 8.0147

y22=y₂ + 0,5/7 p₂₂ = 2,8283 + 0.5-0.5-8.0147 = 4.8320

P32 = 5 e^v2+0,5/' sin(x₂ + 0.5h) - 2y₂₂ = 5 e^1+a25 sin(l + 0,25) - 2-4.8320 = 6.8974

y₃₂ =y₂ + h p₃₂ = 2,8283 + 0.5-6.8974 = 6,2770

p₄₂ = 5 e'^v3 sinx₃ - 2y₃₂ = 5 e^K5 sini,5 - 2-6.2770 = 9.7982

y₃ =yi + ‘4 li (P12 + 2/722 + 2/732 + /7₄₂) = 2,8283 + '/₆-0,5(5,7801 + 2-8,0147 + 2-6,8974 + 9,7982) =

= 6,6119

p\3 = 5 e'^v3 sinx3 - 2y3 = 5 e¹"'¹ sini ,5 - 2-6,6119 = 9.1286 >7,₃ =y₃ + 0,5/7/7,3 = 6.6119 + 0,5-0,5-9,1286 = 8.8940

p₂₃ = 5 e^x3*°’^5h sin(x₃ + 0.5h) - 2y,₃ = 5 e^L5+0-²⁵ sin(l,5 + 0,25) - 2-8.8940 = 10,5242 _V23 = y₃ + 0,5/7 p₂₃ = 6,6119 + 0,5-0,5-10.5242 = 9,2429

p₃₃ = 5 e^r3+0-^5/? sin(x₃ + 0.5h) - 2y₂₃ = 5 e'’⁵⁺⁰’²⁵ sin(l ,5 + 0,25) - 2-9,2429 = 9.8264

>733 = >73 + h p₃₃ = 6,6119 + 0.5-9.8264 = 11.5251

p₄₃ = 5 e'^v4 sinx₄ - 2y₃₃ = 5 e² sin2 - 2-11.5251 = 10.5441

>'4 =y₃ + *4 h (Pi3 + 2/723 + 2p₃₃+p₄₃) = 6,6119 + '4-0,5(9,1286 + 2-10,5242 + 2-9,8264 + 10,5441) = = 11,6430

e) Formuła Adamsa-Bashfortha: y,+, =y, + ^24 {55/(x„ y,) - 59/x/_i, y,_i) + 37J(x,-₂, y_t-₂) - 9 f[x,-₃, >7,-3)] Ponieważ indeksy maleją od /' do i - 3. trzeba znać nie jedną, ale cztery poprzednie wartości - policzone np. metodą Rungego-Kutty IV rzędu.

>4 =>73 + ^;'/₂4{55 /(x₃, >73) - 59 f(x₂, y₂) + 37/(x,. y,) - 9 /(x₀, y₀)] =

= 6.6119 + ⁰‘⁵/₂4-[55(5 e^v3 siat₃ - 2y₃) - 59(5 e^v2 siav₂ - 2>7₂) + 37(5 e^vl sinx, - 2y,) - 9(5 e*° siav₀ - 2y₀)] = = 6.6119 + ⁰'⁵/₂4*[55(5 e¹’⁵ sini.5 - 2-6.6119) - 59(5 e¹ sini - 2-2.8283) + 37(5 e⁰⁵ sin0.5 - 2-0.7728) -

-    9(5 e° sinO - 2-0,3)] = 11,9346

y₅ =y₄ + ^h/₂₄{55 f{x₄, y₄) - 59/(x₃, y₃) + 37/(x₂, y₂) - 9 /(x,, yi)] =

= 6.61 19 +    ⁰⁵/24-[55(5    e'⁴ siav4 - 2y₄) - 59(5 e*³    sinx3 - 2y3) + 37(5 e'² sinx2 - 2y2) - 9(5 e^Al siari -    2y,)] =

= 6.6119 +    °-⁵/₂₄-[55(5    e² sin2 - 2-11.6430) - 59(5 e¹⁵ sini,5 - 2-6.6119) + 37(5 e¹ sini - 2-2,8283)    -

-    9(5 e^{0 5} sin0.5 - 2-0,7728)] = 15,8616

>^;6 =y5 + ^/₂₄{55 /(x₅, ys) - 59 f{x₄, y₄) + 37 f(x₃, y₃) - 9 f[x₂, y₂)] =

= 6.6119 +    ⁰⁵/₂4-[55(5    e^v5 siav₅ - 2y₅) - 59(5 e^v4    sinx-₄ - 2y₄) + 37(5 e^r3 sinx₃ - 2y₃) - 9(5 e^v2 sinx₂ -    2y₂)] =

= 6.6119 +    °'⁵/₂4-[55(5    e²'⁵ sin2.5 - 2-15.8616) -    59(5 e² sin2 - 2-11,9346) + 37(5 e¹'⁵ sini,5 - 2-6,6119) -

-    9(5 e¹ sini - 2-2.8283)] = 15,2820

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