Midterm1Solutions - Ó ÐÙ Ø iÓÒ ×ØÓ id ØeÖÑ AC...

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Unformatted text preview: Ó ÐÙ Ø iÓÒ ×ØÓ id ØeÖÑ AC ÓÔÝÓ fØh eE ÜaÑ i×a ØØh eeÒd G iÚ eÒaÖ×ØÓ Öd e Öd ie ÖeÒ Ø ia ÐeÕÙa Ø iÓÒ iÒÒÓ ÖÑ a ÐfÓ ÖÑ : y prime = f ( t, y ) , Û ecaÒd ÖaÛ Øh ed iÖec Ø iÓÒe ÐdbÝÔ ick iÒga Öb iØÖa ÖÝÔÓ iÒ Ø× ( t, y ) iÒ Øh e ( t, y ) Ô ÐaÒ eaÒdÔ ÐÙgg iÒg Øh e×e iÒ ØÓ f ( t, y ) ØÓÒd Øh ed e Ö iÚa Ø iÚ e y prime a ØØha ØÔÓ iÒ ØF Ó Ö iÒ ×ØaÒ ceÔ ÐÙgg iÒg iÒ Øh eÔÓ iÒ Ø iÒ ØÓ Øh eÖ×Øg iÚ eÒd ie ÖeÒ Ø ia ÐeÕÙa Ø iÓÒ ×aÝ ×Øha ØØh e× ÐÓÔ eÓ fØh e×Ó ÐÙ Ø iÓÒ cÙ ÖÚ ea Ø Ñ Ù ×Øb e h eÓÒ ÐÝd iÖec Ø iÓÒe Ðd ××a Ø i× fÝ iÒg Øh i×a ÖeF igÙ Öe×aÒdB Ýa Ð×Óch eck iÒgeg Û ecaÒ a Ð×Ó ÖÙ ÐeÓÙ ØF igÙ ÖeeÔ ea Ø iÒgØh i×Ô ÖÓce××fÓ ÖØh eÓ Øh e Ög iÚ eÒd ie ÖeÒ Ø ia ÐeÕÙa Ø iÓÒ × bÝ Øe×Ø iÒg k eÝÔÓ iÒ Ø× iÒ Øh e ØÝ Ô ÐaÒ e Û ecaÒ × iÑ iÐa Ö ÐÝÒd Øh ed iÖec Ø iÓÒe Ðd ×ØÓ Øh eÓ Øh e Öd ie ÖeÒ Ø ia Ð eÕÙa Ø iÓÒ ×Ò Øe ÖÑ ×Ó fÒd iÒgØh ecÓ ÖÖe×ÔÓÒd iÒg×Ó ÐÙ Ø iÓÒ cÙ ÖÚ eØÓ Øh e y (0) = 1 ÓÒ ceÛ ekÒÓÛ Øh ed iÖec Ø iÓÒe Ðd × iÑ Ô ÐÝgÓ ØÓ Øh eÔÓ iÒ Ø ÓÒ Øh e ( t, y ) Ô ÐaÒ eaÒdÒdÛ h ich cÙ ÖÚ eÛ iÐÐb eÛ iÐÐ ÐiÒ eÙÔÛ iØh Øh ed iÖec Ø iÓÒe Ðd ØaÒgeÒ ØØ ick ×h e×Ó ÐÙ Ø iÓÒ ×a Öe : Ó ÐÙ Ø iÓÒC Ù ÖÚ eØÓ D ieÖeÒ Ø ia ÐE ÕÙ a Ø iÓÒ D iÖec Ø iÓÒF ie Ðd Øh eÝ = eÒ ØeÖagÙ ÖeÒÙÑ b eÖ4 eÒ ØeÖagÙ Öe Ðe ØØeÖad y prime = 3 y + 1 b y prime = y 2- 5 t- 5 a y prime = y 3- y 2- 6 y 4 c y prime = 3 8 (4 t- 1)( y + 3) + 1 d B ecaÙ ×e f ( t, x ) aÒd iØ×d e Ö iÚa Ø iÚ e×a ÖecÓÒ Ø iÒÙÓÙ ×eÚ e ÖÝÛ h e ÖeØh eÒ iÕÙ eÒ e××h eÓ ÖeÑ hÓ Ðd ×a Ø eÚ e ÖÝÔÓ iÒ Ø iÒ Øh e ØÜ Ô ÐaÒ eh e Öe fÓ Öe×Ó ÐÙ Ø iÓÒ cÙ ÖÚ e×caÒÒÓ Øc ÖÓ ××iÒ ceB Ób ××Ó ÐÙ Ø iÓÒ c ÖÓ ××e× bÓ ØhA ÐiceaÒdC ha Ö Ðie××Ó ÐÙ Ø iÓÒ ×h i× i× iÒ cÓÒ × i×ØeÒ ØÛ iØh Øh e iÖ×Ò Øh eÓ Øh e ÖhaÒdA ÐiceaÒd C ha Ö Ðie××Ó ÐÙ Ø iÓÒ ×dÓÒÓ Øc ÖÓ ×× Øh eÝa Öe ÐiÒ e×Û iØh Øh e×aÑ e× ÐÓÔ e aÒd ×Ó Øh e iÖ×Ó ÐÙ Ø iÓÒ ×dÓÒ Ø Ú iÓ Ða ØeØh eÒ iÕÙ eÒ e××Øh eÓ ÖeÑ iÒ ceeÜac Ø ÐÝ ØÛ Ó×ØÙd eÒ Ø×a ÖecÓ ÖÖec ØiØÑ Ù ×Øb eØha ØA ÐiceaÒd C ha Ö Ðiea ÖecÓ ÖÖec ØaÒdB Ób i×Û ÖÓÒg a eÖecÓgÒ iÞeØh ed ie ÖeÒ Ø ia ÐeÕÙa Ø iÓÒa ×b e iÒgaÙ ØÓÒÓÑ ÓÙ ×× iÒ ce iØha ×fÓ ÖÑ x prime = f ( x ) h e eÕÙ iÐib Ö iÙÑ ×Ó ÐÙ Ø iÓÒ ×cÓ ÖÖe×ÔÓÒd ØÓØh eÞe ÖÓ ×Ó fØh e fÙÒ c Ø iÓÒ f ( x ) = x ( x 2- 4) aÒdh eÒ ceØh e eÕÙ iÐib Ö iÙÑ ×Ó ÐÙ Ø iÓÒ ×a Öe : x 1 (...
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