Өз бетімен шығаруға арналған тапсырмалар:
Исходя из определения устойчивости по Ляпунову, исследовать на устойчивость решения следующих уравнений и систем уравнений:
1.1 ,
1.2 , ![](data:image/png;base64,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)
1.3 ,
1.4 , ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAADwAAAAUCAIAAABeYcl+AAADM0lEQVR4nNWWXSjzURzHvctM1FAbdjdpqJG31VK2xLxfuJgUN6vdiQtKU0pe7tysaD2TS5pWyLZ2sUTSuJBceNtWSLhQyGuI55tTp3+PZefskZ7ne6Hv//y/Oz7/8//9f+fEvb+/R/1viuP9wf7+fl5eXtjY9va2SqWKhIhB3NBOp5MF2uVy/SvQR0dHr6+vLMmYmBi/369QKCKiCiM+6MXFxfz8fHo5NzfX09ODxxgeHjYajcIkYh6Phxd6YWGht7f38PDw6xgf9NbWVmVlJfG7u7sTExMYeXx8bGxsRDGUlJTQZFZWlsPhYJ/5/v5+dHRUKpUGAoGwYT7og4OD9PR04i0WS19fX2ZmJvzAwMDk5OTU1BRNZmRkBINB9pmTk5NHRkZgurq6IoH2+Xw1NTUGg8FqteJyfX29tbX17OwM/vT0NDU1lcRWV1cHBweJLysro54oLS3t+PiYHZpLIaArKirm5+ebm5vHx8exAEtLS9PT0+TW3d1dQkIC8ScnJyAjXiKR4HmEk8THx9/e3sJ4vd7q6uqQ/zs2Nvbl5eV7oKGqqqqCggKbzbazs9Pe3q7Vask4Ko9Co5QTExOJF4lEuBTOgFsPDw8wOp3u7e0tAjJuaMhsNre0tGxsbBQXF9PB6Oho6pOSkp6envA36uMNEEOFjVYY/l6FhsaKzs7O4iO7uLgQjqNanp+fyQLn5ORcXV0R1svLS7QLYRIxhH8OGh/Q2NjY0NAQPi980XV1dfSWWCym0Gq1Gv1OJpPBb25uIiycBMVKoH+ipsHhdrvRgLGldXR09Pf3Ly8vo8TJ3ezs7Jubm5SUFHiTydTd3V1aWopqxrMJ+x10fX2NcNRf17RGo1lbWwsD3dDQAEoQw6O7tbW1NTU1Yeerra3FCE4dqARCg6Xt7OwsLCxEGDtieXm5cB7EWI4oVOfn52iye3t78ChLpVJpt9vJJvBZf0KTfkz160P0EnseWhs9CZk+FHJexIqKitihsReurKx8Hv+8zCGgvxZWfWZmBm8jbBKbPN4S1+Ts4oNGYdDe/LXQzuVyeURI4cV9ntbr9Syx+vp6fhhWcUMznjZzc3P5YVj1G9D5O8lWd93YAAAAAElFTkSuQmCC)
1.5
1.6
№2 практикалық сабақ
Сызықты дифференциалдық жүйелердің орнықтылғы туралы жалпы теоремалар.
Мысалдар.
1. Скалярлық
теңдеуінің барлық шешімдері:
шенелмеген. Олар шешімінің айналасына шоғырланған және оған кезде шексіз ұмтылады. Сондықтан шешімі жай ғана асимптотикалық орнықты. Сонымен қатар берілген теңдеудің барлық шешімі шенелмеген.
2. Скалярлы
теңдеуін қарастырайық. Интегралдасақ,
, егер
, егер
Әлбетте теңдеудің барлық шешімдері -де шенелген (9-сызба). Бірақ нөлдік шешім кезде орнықсыз, себебі ![](data:image/png;base64,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)
![](data:image/png;base64,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) ![](data:image/png;base64,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)
9-сызба
Достарыңызбен бөлісу: |