1-mavzu. Mavzu. Differensial tenglama. Asosiy tushuncha va ta’riflar. Birinchi tartibli differensial tenglamalar uchun Koshi masalasi

3. Laplas tasvirining ba’zi xossalari.

1. 
   Chiziqlilik xossasi. Agar F_k(t) -¸® f_k(t) (k=1,2,3,...,n) bo’lib, s_k lar o’zgarmaslar bo’lsa , u holda c_kF_k(t) -¸® c_kf_k(t) munosabat o’rinli bo’ladi.
   

2. O’xshashlik teoremasi. Agar a>0 va F(t) -¸®f(t) bo’lsa , u holda F( ) -¸® f(at) bo’ladi.

3. Originalning kechikish teoremasi. Agar l>0 bo’lsa, u holda F(t) -¸® f(t) dan f(t-l) -¸® e^-ltF(t) kelib chiqadi.

4. Tasvirning sinish teoremasi. Agar F(t) -¸® f(t) bo’lsa, u holda istalgan l uchun e^-ltf(t) -¸® F(t+l) kelib chiqadi.

5. Originalni differensiallash. Agar f(t) funksiya [0,¥] da uzluksiz, differensiallanuvchi va f ¢(t) hosila tasvir mavjudligining 1,2,3 shartlarini qanoatlantirib, F(t) -¸® f(t) bo’lsa, quyidagilar o’rinli bo’ladi.

a) pF(p)-f(0) -¸® f ¢(t) Xususiy holda f(0)=0 bo’lsa pF(p) -¸® f ¢(t) bo’ladi.

2. 
   Agar fⁿ(t) mavjud bo’lsa va tasvir mavjudligining shartlarini qanoatlantirsa
   

f⁽ⁿ⁾(t) -¸® tⁿF(t)-[ t^n-1f(0)+ t^n-2f¢(0)+ t^n-3f¢¢(0) +...+ tf^(n-2)(0)+f^(n-1)(0) ],

agar f(0)=f ¢(0)=...=f^(n-1)(0)=0 bo’lsa, f⁽ⁿ⁾(t) -¸®tⁿF(t) bo’ladi.

6. Originalni integrallash. Agar F(p) -¸® f (t) bo’lsa,

f(t)dt ning tasviri f(t)dt -¸® bo’ladi.

7. Tasvirni integrallash. Agar F(p) -¸® f (t) bo’lsa, F(t)dt -¸® bo’ladi.

8. Tasvirni differensiallash. Agar F(p) -¸® f (t) bo’lsa , u holda

1. 
   F¢(p) -¸® -tf(t) ; b) F⁽ⁿ⁾(p) -¸® (-1)ⁿtⁿf(t) bo’ladi.
   

Misol. f(t)=sinat funksiya tasvirini topamiz: F(p)= sinat×e^-ptdt=1/a×1/[(p/a)²+1] =a/(p²+a²) Demak sinat¬¸-a/(p²+a²) (5)

Shunga o’hshash f(t)=cosat funksiyani tasviri quyidagicha bo’ladi: cosat¬¸-p/(p²+a²) (6)

Misol. f(t)=3sin4t-2cos5t funksiya tasviri topilsin.

Yechish: (5) va (6) dan Líf(t)ý=3×4/(p²+16)-2×p/(p²+25)=12/(p²+16)-2p/(p²+25).

Misol. F(t)=5/(t²+4)+20t/(t²+9) tasvir funksiya berilganda boshlangich funksiyani toping.

Yechish: F(t)=5/2×2/(p²+4)+20×p/(p²+9) -¸®5/2×sin2t+20cos3t=f(t)

Demak f(t)=5/2×sin2t+20×cos3t.

Misol. f(t)=e^-at funksiyani tasviri topilsin. F(t) = e^-pte^-atdt == e^-(p+a)tdt=1/(p+a) (7)

Misol. f(t)=e^at funksiyani tasviri topilsin. F(t) = e^-pte^atdt == e^-(p-a)tdt=1/(p-a) (8)

Misol. f(t)=shat=1/2×(e^at-e^-at) funksiyani tasviri topilsin.

(7) va (8) dan 1/2×(e^at-e^-at) ¬¸-1/2×[1/(1-p)-1/(p+a)]=a/(p²-a²)

Demak F(t)=a/(p²-a²) yoki F(t) -¸®f(t) ya’ni shat¬¸-a/(p²-a²) (9)

Shunga o’xshash chat=(e^at+e^-at)/2 funksiya tasviri chat ¬¸- p/(p²-a²) (10)

Misol. F(t)= 7/(p²+10p+41) tasvir funksiyadan boshlangich funksiya topilsin.

Yechish: F(t)= 7/(p²+10p+41)= 7×4/[4×((p+5)²+4²)] Demak

7×4/[4×((p+5)²+4²) -¸®7/4×e^-5tsin4t. yoki F(t)-¸®7/4×e^-5tsin4t=f(t)

Misol. F(t)=(p+3)/(p²+2p+10) tasvir funksiyadan boshlang’ich

funksiya topilsin.

F(t)=(p+3)/(p²+2p+10)=[(p+1)+2]/[(p+1)²+9]=(p+1)/[(p+1)²+3²]+2/[(p+1)²+3²]=

=(p+1)/[(p+1)²+3²]+2/3×3/[(p+1)²+3²]

Demak F(t) -¸®e^-tcos3t+2/3×e^-tsin3t f(t)= e^-tcos3t+2/3×e^-tsin3t.