Math 2320 Differential Equations Chapter 3.2 Applications of First Order Linear Differential Equations

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1 Mah 3 Differenial Equaions Chaper 3. Applicaions of Firs Order Linear Differenial Equaions Mixing roblems: The basic mixing problem consiss of a funcion ha represens he amoun of a subsance in a ank a ime, an inpu rae a which he subsance eners he ank, and an oupu rae a which he subsance leaves he ank. Since he derivaive represens he rae of change in he amoun of he subsance in he ank a ime we have: dx inpu rae oupu rae The rae of change for boh inpu and oupu is he produc of he flow rae (volume/ime) and he concenraion (amoun/volume).. A ank conains liers of fluid in which 3 grams of sal is dissolved. Brine conaining gram of sal per lier is hen pumped ino he ank a a rae of 4 L/min; he well mixed soluion is pumped ou a he same rae. Find he number 5 5 e A 4e 5 5 e A e C 5 A ( ) Ce A() 3 A of grams of sal in he ank a ime. da inpu rae oupu rae = gram 4 A liers 4L lier min ue L min ue A 4 5 da A inegraing facor e e e A 4e A( ) 7 5 e

2 . A large ank is parially filled wih gallons of fluid in which pounds of sal is dissolved. Brine conaining pound of sal per gallon is pumped ino he ank a a rae of 6 gal/min. The well mixed soluion is hen pumped ou a a slower rae of 4 gal/min. Find he number of pounds of sal in he ank afer 3 minues. da inpu rae oupu rae lb 6gallons A 4gallons = gallon min ue gallons min ue A 3 5 da A ln5 5 3 inegraing facor e e 5 5 d 5 A 35 5 A A5 C 5 C 5 A () 5 C 5 A() C, 5,5 A A 3 8, lb

3 3. A niric acid soluion flows a a consan rae of 6 L/min ino a large ank ha iniially held L of a.5% niric acid soluion. The soluion inside he ank is kep well sirred and flows ou of he ank a a rae of 8 L/min. If he soluion enering he ank is % niric acid, deermine he volume of niric acid in he ank a ime minues. When will he percenage of niric acid in he ank reach %? da inpu rae oupu rae A.68 A. da A ln. Inegraing facor e e d A. A.. C A C A. C A().5(). C.9 A..9 A minues

4 opulaion Dynamics: One of he earlies aemps o model human populaion growh by means of mahemaics was by he English economis Thomas Malhus. The idea behind he Malhusian model is he assumpion ha he rae a which he populaion of a communiy grows a a cerain ime is direcly proporional o he oal populaion of he communiy a ha ime. If denoes he populaion a ime, hen his assumpion can be expressed as d k, () where k is he consan of proporionaliy and is he iniial populaion 4. The populaion of a own grows a a rae proporional o he populaion presen a ime. The iniial populaion of 5 increases by 5% in years. Wha will be he populaion in 3 years? d k, () 5 Solve by separaion of variables d k d k ln k c ln kc e e k Ce When = he populaion is 5 Ce k 5 Ce C 5 k k 5 e When =, he populaion of 5 decreases by 5% o 45 k 45 5e 45 5 e 45 ln k 5 45 ln 5 k.6 (.6)(3) 5e 33 people

5 5. The populaion of a communiy is known o increase a a rae proporional o he number of people presen a ime. If he populaion has doubled in 5 years, how long will i ake o riple? To quadruple? d k, () d k d k ln k c k Ce k e In 5 years he iniial populaion doubles e e ln e ln ln ln k e.39 3 e.39 ln 3 ln e ln years.39 ln years The radioacive isoope of lead, b 9, decays a a rae proporional o he amoun presen a ime and has a half life of 3.3 hours. If gram of lead is presen iniially, how long will i ake for 9% of he lead o decay? d k, () k e.5 e 3.3k 3.3k ln.5 ln.5 k. 3.3 e... ln. ln..96. hours

6 7. The iniial mass of a cerain species of fish was 7 million ons. The mass of he fish, if lef alone, would increase a a rae proporional o is mass, wih a proporionaliy consan of /yr. However, commercial fishing removes he fish populaion a a consan rae of 5 million ons per year. When will all he fish be gone? If he fishing rae is changed so ha he mass of fish remains consan, wha should ha rae be?