CCF20090601014

y₂o ~ jo + 0,5 h P20 — 0,3 + 0,5-0,5-1,2884 — 0,6221

P30 = 5 e^x0+0’^5h _sin(x₀ + 0,5h) - 2y₂₀ = 5 e⁰⁺⁰-²⁵ sin(0 + 0,25) - 2-0,6221 = 0,3442

J30 =yo + hp₃₀ = 0,3 + 0,5-0,3442 = 0,4721

pąo = 5 e^Al simą - 2j₃o = 5 e⁰"'’ sin0.5 - 2-0,4721 = 3.0080

ji = j₀ + ‘/₆ /i (p_to + 2p₂₀ + 2p₃₀ + p₄₀) = 0,3 + V₆-0,5(-0,6 + 2-1,2884 + 2-0,3442 + 3,0080) = 0,7728

Pu =5 e^xl simą - 2y, = 5 e⁰’⁵ sin0.5 - 2-0,7728 = 2,4067 yn=yi+ 0,5/2 pu = 0,7728 + 0,5-0,5-2,4067 = 1,3744

p₂\ = 5 e^xl+0’^5/? sin(x, + 0,5h) - 2y_u = 5 e°’⁵⁺⁰’²⁵ sin(0,5 + 0,25) - 2-1,3744= 4,4663 J21 = y\ + 0,5/7 p₂\ = 0.7728 + 0,5-0,5-4,4663 = 1,8893

p₃\ = 5 e^xl+0’^5h _Sin(x, + 0,5h) - 2j₂₎ = 5 e⁰’^5+a25 sin(0.5 + 0,25) - 2-1,8893 = 3,4365

J31 = ji + hpu = 0,7728 + 0,5-3,4365 = 2,4910

7?4i = 5 e'² siriX2 - 2j₃i = 5 e¹ sini - 2-2.4910 = 6,4548

j₂ = Ji + V₆ h(p\\+ 2p₂i + 2p₃i + P4i) = 0,7728 + ‘/₆-0,5(2,4067 + 2-4,4663 + 2-3,4365 + 6,4548) =

= 2,8283

pn = 5 e'² sinx₂ - 2j₂ - 5 e¹ sini - 2-2,8283 = 5,7801

J12 = J2 + 0,5/2 _Pi ₂ = 2,8283 + 0,5-0,5-5,7801 = 4,2734

p₂₂ = 5 e^x2+0'^5/' sin(x₂ + 0,5/2) - 2j,₂ = 5 eⁱ⁺⁰’²⁵ sin(l + 0,25) - 2-4,2734 = 8,0147

J22 = J₂ + 0,5/2 p₂₂ = 2,8283 + 0,5-0,5-8,0147 = 4.8320

p₃2 = 5 e^x2+0-^5h _Sin(x₂ + 0,5h) - 2y₂₂ = 5 e^l+a25 sin(l + 0,25) - 2-4,8320 = 6,8974

J32 =yi + hp₃2 = 2,8283 + 0,5-6,8974 = 6,2770

p₄₂ = 5 e*³ sinx₃ - 2j₃₂ = 5 e¹'⁵ sini ,5 - 2-6,2770 = 9,7982

J3 = J2 + '/(, h (Pu + 2p₂₂ + 2p_n + Pu) = 2,8283 + V₆-0,5(5,7801 + 2-8,0147 + 2-6,8974 + 9,7982) =

= 6,6119

pu = 5 e^x3 sinx₃ - 2j₃ = 5 e^K5 sin 1,5 - 2-6,6119 = 9.1286 J13 = J3 + 0,5 h p\₃ = 6,6119 + 0,5-0,5-9,1286 = 8,8940

P23 = 5 e^x3+0’^5/l sin(x₃ + 0,5/2) - 2j,₃ = 5 e'-⁵⁺⁰'²⁵ sin(l,5 + 0.25) - 2-8,8940 = 10,5242 J23 = J3 + 0,5/2 p₂₃ = 6,6119 + 0,5-0,5-10,5242 = 9,2429

P33 = 5 e^x3+0'^5/' sin(x₃ + 0,5h) - 2j₂₃ = 5 e'-⁵⁺⁰’²⁵ sin(l,5 + 0,25) - 2-9,2429 = 9,8264

J33 = J₃ + h_P33 = 6,6119 + 0,5-9,8264 = 11,5251

P43 = 5 e'⁴ sinx4 - 2j33 = 5e² sin2 - 2-11,5251 = 10,5441

J4 = J3 + '/₆ h (Pu + 2/723 + 2pu + /7₄₃) = 6,6119 + '/₆-0,5(9,1286 + 2-10,5242 + 2-9,8264 + 10,5441) = = 11,6430

e) Formuła Adamsa-Bashfortha: y_i+\ = y, + ^h/₂4-[55J(x_i, y,) - 59 /(x,_i, j_;-i) + 37 /(x,_2, y,-i) - 9y(x,-₃₎ j,_₃)] Ponieważ indeksy maleją od i do i - 3, trzeba znać nie jedną, ale cztery poprzednie wartości - policzone np. metodą Rungego-Kutty IV rzędu.

J4 = J3 + V[55./tx₃, >’3) - 59 /(x₂, y₂) + 37./(xi, j]) - 9 f[x₀, jo)] =

= 6,6119 + °"724-[55(5 e^AJ siax₃ - 2j₃) - 59(5 e^x2 sinx2 - 2j₂) + 37(5 e^Al sinxi - 2ji) - 9(5 e^x0 sinxo - 2jo)] = = 6,6119 + °-⁵/₂₄-[55(5 e¹'⁵ sini,5 - 2-6,6119) - 59(5 e¹ sini - 2-2,8283) + 37(5 e⁰’⁵ sin0,5 - 2-0,7728) -

-    9(5 e° sinO - 2-0,3)] = 11,9346

Js = J4 + ^;7₂₄ -[55 /(x₄. j₄) - 59 /(x₃, j₃) + 37 /(x₂, j₂) - 9 f[x_h ji)] =

= 6.6119 + °'⁵/₂₄-[55(5    e^x4 sinx₄ - 2j₄) - 59(5 e^x3 sinx₃ - 2j₃) + 37(5 e^x2 sinx₂ - 2j₂)    - 9(5 e^xl sinx, - 2j,)]    =

= 6,6119 + °'⁵/₂₄-[55(5    e² sin2 - 2-11,6430) - 59(5 e¹'⁵ sini,5 - 2-6,6119) + 37(5 e¹    sini - 2-2,8283) -

-    9(5 e⁰-⁵ sin0.5 - 2-0,7728)] = 15,8616

J6 = J5 + V₂₄-[55./(x₅₎ Js) - 59 /(x₄, j₄) + 37 /(x₃, j₃) - 9f[x₂, j₂)] =

= 6.6119 + ^0,5/₂4-[55(5    e^x5 sinxj - 2j₅) - 59(5 e^x4 sinx₄ - 2j4) + 37(5 e^XJ sinx₃ - 2j₃)    - 9(5 e'² sinx₂ - 2j₂)]    =

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

-    9(5 e¹ sini - 2-2,8283)] = 15,2820

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