7.5-bo'lim: Laplas o'zgarishlari
Haqiqatan ham bu bo'limda unchalik ko'p narsa yo'q. Bu erda biz qiladigan narsa - 3 uchun Laplace o'zgarishlaridan foydalangan holda tezkor misolrd differensial tenglamani tartiblang, shuning uchun biz tartibi 2 dan katta bo'lgan differentsial tenglama uchun kamida bitta muammoni ishladik deb aytishimiz mumkin.
Biz bilgan hamma narsa Laplas o'zgarishlari bo'lim hali ham amal qiladi. Bu erda bizga kerak bo'lgan yagona yangi narsa - bu uchinchi hosilaning Laplas konvertatsiyasi. Biz buni dan olishimiz mumkin umumiy formula Biz birinchi marta IVP ni Laplas transformatsiyasi bilan hal qilishni boshlaganimizda bergan edik. Mana bu formula,
\[\mathcal{L}\left\{ {y'''} \right\} = {s^3}Y\left( s \right) - {s^2}y\left( 0 \right) - sy'\left( 0 \right) - y''\left( 0 \right)\]Mana bu bo'lim uchun misol.
Yechim
Har doimgidek, biz birinchi navbatda Heaviside funktsiyasi bilan ko'paytiriladigan funktsiya to'g'ri o'zgartirilganligiga ishonch hosil qilishimiz kerak.
\[y''' - 4y'' = 4t + 3{u_6}\left( t \right){{\bf{e}}^{ - 5\left( {t - 6} \right)}}\]U to'g'ri o'zgartirildi va biz \({{\bf{e}}^{ - 5t}}\) ni o'zgartirayotganimizni ko'rishimiz mumkin. Hozir qilishimiz kerak bo'lgan narsa hamma narsani Laplas konvertatsiyasini olish, boshlang'ich shartlarni ulash va \(Y\left( s \right)\) ni hal qilish. Bularning barchasini bajarish,
\[\begin{align*}{s^3}Y\left( s \right) - {s^2}y\left( 0 \right) - sy'\left( 0 \right) - y''\left( 0 \right) - 4\left( {{s^2}Y\left( s \right) - sy\left( 0 \right) - y'\left( 0 \right)} \right) & = \frac{4}{{{s^2}}} + \frac{{3{{\bf{e}}^{ - 6s}}}}{{s + 5}}\\ \hspace{0.25in}\hspace{0.25in}\left( {{s^3} - 4{s^2}} \right)Y\left( s \right) + 3{s^2} - 13s & = \frac{4}{{{s^2}}} + \frac{{3{{\bf{e}}^{ - 6s}}}}{{s + 5}}\\ \hspace{0.25in}\hspace{0.25in}\left( {{s^3} - 4{s^2}} \right)Y\left( s \right) & = \frac{4}{{{s^2}}} - 3{s^2} + 13s + \frac{{3{{\bf{e}}^{ - 6s}}}}{{s + 5}}\\ \hspace{0.25in}\hspace{0.25in}\left( {{s^3} - 4{s^2}} \right)Y\left( s \right) & = \frac{{4 - 3{s^4} + 13{s^3}}}{{{s^2}}} + \frac{{3{{\bf{e}}^{ - 6s}}}}{{s + 5}}\\ \hspace{0.25in}\hspace{0.25in}\hspace{0.25in}\hspace{0.25in}\,\,\,\,\,\,\,\,Y\left( s \right) & = \frac{{4 - 3{s^4} + 13{s^3}}}{{{s^4}\left( {s - 4} \right)}} + \frac{{3{{\bf{e}}^{ - 6s}}}}{{{s^2}\left( {s - 4} \right)\left( {s + 5} \right)}}\\ \hspace{0.25in}\hspace{0.25in}\,\,\,\,\,\,\,\,\,\,\,Y\left( s \right) & = F\left( s \right) + 3{{\bf{e}}^{ - 6s}}G\left( s \right)\end{align*}\]Endi biz qisman kasr va \(F(s)\) va \(G(s)\) teskari aylantirishimiz kerak. Tafsilotlarni tekshirishni sizga qoldiramiz.
\[\begin{align*}F\left( s \right) & = \frac{{4 - 3{s^4} + 13{s^3}}}{{{s^4}\left( {s - 4} \right)}} = \frac{{\frac{{17}}{{64}}}}{{s - 4}} - \frac{{\frac{{209}}{{64}}}}{s} - \frac{{\frac{1}{{16}}}}{{{s^2}}} - \frac{{\frac{1}{4}\left( {\frac{{2!}}{{2!}}} \right)}}{{{s^3}}} - \frac{{1\left( {\frac{{3!}}{{3!}}} \right)}}{{{s^4}}}\\ f\left( t \right) & = \frac{{17}}{{64}}{{\bf{e}}^{4t}} - \frac{{209}}{{64}} - \frac{1}{{16}}t - \frac{1}{8}{t^2} - \frac{1}{6}{t^3}\end{align*}\]\[\begin{align*}G\left( s \right) & = \frac{1}{{{s^2}\left( {s - 4} \right)\left( {s + 5} \right)}} = \frac{{\frac{1}{{144}}}}{{s - 4}} - \frac{{\frac{1}{{225}}}}{{s + 5}} - \frac{{\frac{1}{{400}}}}{s} - \frac{{\frac{1}{{20}}}}{{{s^2}}}\\ g\left( t \right) & = \frac{1}{{144}}{{\bf{e}}^{4t}} - \frac{1}{{225}}{{\bf{e}}^{ - 5t}} - \frac{1}{{400}} - \frac{1}{{20}}t\end{align*}\]
OK, endi biz differentsial tenglamaning yechimini olamiz. Biz olgan transformatsiyadan boshlab,
\[Y\left( s \right) = F\left( s \right) + 3{{\bf{e}}^{ - 6s}}G\left( s \right)\hspace{0.25in}\hspace{0.25in} \Rightarrow \hspace{0.25in}\hspace{0.25in}y\left( t \right) = f\left( t \right) + 3{u_6}\left( t \right)g\left( {t - 6} \right)\]Bu erda \(f(t)\) va \(g(t)\) yuqorida ko'rsatilgan funktsiyalardir.
To‘g‘ri, 2 dan katta tartibli differensial tenglamaga ega bo‘lgan bitta Laplas o‘zgartirish misoli bor. Ko‘rib turganingizdek, qisman kasr ishi (biz buni bu yerda ko‘rsatmadik) chalkashroq va murakkabroq bo‘lishidan tashqari ish bir xilda ko‘rinib turibdi.