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205sp11-ex1-solutions - A5G4:EXA dy dx = xy 2 y(0 = 1 ÔÓ...

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Unformatted text preview: A5G4 :EXA# dy dx = xy 2 y (0) = 1 ÔÓ iÒ Ø× Ó ÐÙ Ø iÓÒ : h i×i×a ×eÔa Öab ÐeeÕÙa Ø iÓÒ 1 y 2 dy = xdx. Ò Øeg Öa ØebÓ Øh × id e×aÒd- 1 y = 1 2 x 2 + C, Ó Ö y =- 1 1 2 x 2 + C . h e iÒ iØ ia ÐÓÒd iØ iÓÒg iÚ e× 1 = y (0) =- 1 1 2 2 + C =- 1 C , ×Ó C =- 1 Ó Ö Ñ Ù ÐØ iÔ ÐÝ iÒg ØÓÔaÒdbÓ ØØÓÑ bÝ y = 2 2- x 2 . y ′ + 1 x y = √ x 2- 1 y (1) = 2 5ÔÓ iÒ Ø× Ó ÐÙ Ø iÓÒ : A ÐiÒ ea ÖeÕÙa Ø iÓÒ iÒ ×ØaÒda Öd fÓ ÖÑ p = 1 /x P = ln( x ) aÒd Øh e iÒ Øeg Öa Ø iÒg faØÓ Ö μ = x ×Ó ( μy ) ′ = ( xy ′ + y ) = x √ x 2- 1 Ò Øeg Öa ØebÓ Øh × id e×aÒd xy = 2 3 1 2 ( x 2- 1) 3 / 2 + C, ÝÓÙÒ eed ØÓdÓa u ×Ùb ×Ø iØÙ Ø iÓÒ u = x 2- 1 fÓ ÖØha Ø iÒ Øeg Öa Ð Ó Ö y = 1 3 x ( x 2- 1) 3 / 2 + C x . h e iÒ iØ ia ÐÓÒd iØ iÓÒ iÑ Ô Ðie×Øha Ø C = 2 , y = 1 3 x ( x 2- 1) 3 / 2 + 2 x . A ØaÒkÛ hÓ ×eÚÓ ÐÙÑ e i× 40 L iÒ iØ ia ÐÐÝÓÒ Øa iÒ × 20 L Ó f×a ÐØÛ a Øe Öa ØaÓÒeÒ ØÖa Ø iÓÒÓ f 5 g/L A ×Ó ÐÙ Ø iÓÒÓÒ Øa iÒ iÒg 10 g/L Ó f×a ÐØ i×ÔÙÑ Ô ed iÒ ØÓ Øh eØaÒka Øa Öa ØeÓ f 4 L/min aÒd Øh eÛ e ÐÐ×Ø iÖÖedÑ iÜ ØÙ ÖeÓÛ ×ÓÙ Øa Øa Öa ØeÓ f 3 L/min ÓÛÑ Ùh ×a ÐØÛ iÐÐb e iÒ Øh e ØaÒkÛ h eÒ iØ×Øa ÖØ×ØÓÓÚ e ÖÓÛ ? 5ÔÓ iÒ Ø× A5G4 :EXA# Ó ÐÙ Ø iÓÒ : f A ( t ) i×Øh eaÑ ÓÙÒ ØÓ f×a ÐØ iÒ Øh eØaÒka ØØ iÑ e t Øh eÒ Øh ed ie ÖeÒ Ø ia Ð eÕÙa Ø iÓÒÑ Ód e ÐiÒg Øh i×× iØÙa Ø iÓÒ i× A ′ = 40- 3 A V , × iÒeØh e Öe i× 4 Ñ iÒÓÑ iÒg iÒÛ iØhÓÒeÒ ØÖa Ø iÓÒ 10...
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