1. Muammo: Berilgan tizimning yechimini topish kerak: $$9y'' + 4y''' + 2y' + 8y = 2u'' + 8u'' + 4u' + 2u$$, bu yerda $y$ chiqish, $u$ kirish. 2. Avvalo, tenglamani soddalashtiramiz. O'ng tomonda $2u'' + 8u'' = 10u''$ bo'ladi, shuning uchun tenglama: $$9y'' + 4y''' + 2y' + 8y = 10u'' + 4u' + 2u$$ 3. Tizimni yechish uchun Laplas transformatsiyasidan foydalanamiz. $Y(s)$ va $U(s)$ mos ravishda $y(t)$ va $u(t)$ ning Laplas transformatsiyalari. 4. Har bir hosilani Laplasga o'tkazamiz: $$\mathcal{L}\{y'\} = sY(s) - y(0), \quad \mathcal{L}\{y''\} = s^2Y(s) - sy(0) - y'(0), \quad \mathcal{L}\{y'''\} = s^3Y(s) - s^2y(0) - sy'(0) - y''(0)$$ 5. Faraz qilamizki, boshlang'ich shartlar nolga teng: $y(0) = y'(0) = y''(0) = 0$. 6. Tenglamaning Laplas ko'rinishi: $$9s^2Y(s) + 4s^3Y(s) + 2sY(s) + 8Y(s) = 10s^2U(s) + 4sU(s) + 2U(s)$$ 7. $Y(s)$ ni ajratamiz: $$Y(s)(4s^3 + 9s^2 + 2s + 8) = U(s)(10s^2 + 4s + 2)$$ 8. O'tkazish funksiyasi (transfer function) quyidagicha bo'ladi: $$H(s) = \frac{Y(s)}{U(s)} = \frac{10s^2 + 4s + 2}{4s^3 + 9s^2 + 2s + 8}$$ 9. Bu o'tkazish funksiyasidan tizimning javobini topish uchun $U(s)$ ni bilish kerak. Agar $u(t)$ ma'lum bo'lsa, $Y(s) = H(s)U(s)$ orqali yechim topiladi. 10. Agar $u(t)$ impuls yoki boshqa oddiy funksiya bo'lsa, invers Laplas yordamida $y(t)$ ni topish mumkin. Natija: Tizimning yechimi Laplas transformatsiyasi yordamida $Y(s) = H(s)U(s)$ ko'rinishida ifodalanadi, bunda $$H(s) = \frac{10s^2 + 4s + 2}{4s^3 + 9s^2 + 2s + 8}$$