Fizika-matematika fakulteti

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Mashq l ar

1.4. Integralning ikkala chegarasi o‘zgaruvchi bo‘lgan integral tenglamalar

Oldingi paragraflardan ma‟lumki, Fredgolm tenglamasidagi integralning ikkala chegarasi ham o„zgarmas, Volter tenglamasida esa integralning chegaralaridan biri o„zgaruvchi bo„ladi.

Tenglamadagi integralning ikkala chegarasi ham o„zgaruvchi bo„lishi mumkin. Quydagi tenglamalar ana shunday tenglamalardandir:
x
^u⁽^x⁾^^^f⁽^x⁾^^^^K⁽^x^,^t⁾^u⁽^t⁾^d^t^,^(1.²⁸⁾x
bunda 0 - o„zgarmas haqiqiy son;
x
^u⁽^x⁾^^^f⁽^x⁾^^^^K⁽^x^,^t⁾^u⁽^t⁾^d^t^,(1.29) px
_t_eng_l_a_m_a_l_arda_p_-_o_„_zga_r_m_as_son(0 p 1).
_Yuq_o_r_i_dag_i_(1.₂₈₎_t_eng_l_a_m_adaa x b, x t x _debtushunmoq kerak; (1.29) tenglamada axb va t esa px bilan x orasida
o„zgaradi, ya‟ni
_agar_x__₀_bo_„l_sa,pxtx, _agar_x__₀_bo_„l_sa,xtpx.
Mana shu tur integral tenglamalardagi noma‟lum funksiyalar ham ikki argumentli bo„lib qolishi mumkin. Masalan,

20
^x ^y
^u⁽^x^,^y⁾^^^f⁽^x^,^y⁾^^^^K⁽^x^,^y^,^t¹^,^t²^,⁾^u⁽^t¹^,^t²⁾^d^t¹^d^t²^,(1.30) xx
_b_un_da0 _v_a0 ^o_„_zgar_m_a_s_ha_q_i_q_i_y_son_l_a_r_;_yok_i
^x ^y
^u⁽^x^,^y⁾^^^f⁽^x^,^y⁾^^^^K⁽^x^,^y^,^t¹^,^t²^,⁾^u⁽^t¹^,^t²⁾^d^t¹^d^t²^,(1.31) px py
bunda ⁰^^^p^¹va ⁰^^^q^¹o„zgarmas haqiqiy sonlar.

(1.28) va (1.29) tenglamalarga o„xshash tenglamalar konkret masalalarni hal qilishda uchraydi. Bunday umumiyroq ko„rinishda quyidagicha yozish mumkin.
b(x)
^u⁽^x⁾^^^f⁽x⁾^^^^K⁽^x^,^t⁾^u⁽^t⁾^d^t^,a(x)
texnikaga doir tenglamalarni

(1.32)

Bundagi integrallarning ikkala chegarasi ham x ning funksiyalaridir. Ularni yechishning har xil usullarini kelgusi boblarda qarab chiqamiz.

Mashqlar

Quyidagi ikkala chegarasi o„zgaruvchi bo„lgan integral tenglamalarni yeching:

x
1. ^u⁽^x⁾^¹^^^^ekt^u⁽^t⁾^d^t^;bu yerda ^^^^⁰, ^k^^⁰o„zgarmas sonlar. x
x
_2.^u⁽^x⁾^^^x^¹^^^^^ekt^u⁽^t⁾^d^t^;4^x
x
^3.^u⁽^x⁾^^a^x^^^⁽^x2 ^^t2 ⁾^u⁽^t⁾^d^t^;^a^^⁰^,⁰^^^p^¹^.px
qx
4. ^u⁽^x⁾^^^a^x2 ^^^^u⁽^t⁾^d^t^;⁰^^^p^¹^,⁰^^^p^¹^.px

x
_5.^u⁽^x^,^y⁾^¹^^^x
y
^eh(t1t2 )^u⁽^t¹^,^t²⁾^d^t¹^d^t²^,_b_u_yer_d_a____₀_,_____0,x

y
^k0 – o„zgarmas sonlar.

x
6. ^u⁽^x^,^y⁾^^^x^y^^^^x

¹^u⁽^t¹^,^t²⁾^d^t¹^d^t²^,0 < p, q < 1. x¹²

x
y

1 2
7. ^u⁽^x^,^y⁾^^^a^x^y^^^^⁽^x2 ^y2 ^^t²^t¹⁾^u⁽^t¹^,^t²⁾^d^t¹^d^t²^,0 < p, q < 1. xx

y

x
8. ^u⁽^x^,^y⁾^^^a^^^^¹^u

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