ln y t 2 t c where c is an arbitrary real constant

Transcription

1 SOLUTION TO THE PROBLEM.A y y subjec o condiion y 0 8 We recognize is as a linear firs order differenial equaion wi consan coefficiens. Firs we sall find e general soluion, and en we sall find one a saisfies e given condiion. y y We recognize e cain rule, y dy d ln y, and plug is ino e above. y d y d d ln y d Inegraing bo sides wi respec o ln y c were c is an arbirary real consan Exponeniaing bo sides: exp exp y c In case y 0 we ave y expc exp In case y 0 we ave y expc exp Since exp c 0, we can wrie e general soluion wiou e absolue value as exp y were is an arbirary real consan y exp 0 8 Plugging in e condiion Te soluion saisfying e given condiion is y 8exp.

2 SOLUTION TO THE PROBLEM.B 5 y y subjec o condiion y 0 We recognize is as a linear firs order differenial equaion wi consan coefficiens. y y y 0.5 y y 0.5 We recognize e cain rule, y dy d ln y 0.5. y 0.5 d y 0.5 d d ln y 0.5 d Inegraing bo sides wi respec o ln y 0.5 c were c is an arbirary real consan Exponeniaing bo sides: 0.5 exp exp y c In case y we ave y c exp exp 0.5 In case y we ave y c exp exp 0.5 Since exp c 0, we can wrie e general soluion wiou e absolue values as y exp 0.5 were is an arbirary real consan y exp Plugging in e condiion Te soluion saisfying e given condiion is y exp 0.5.

3 SOLUTION TO THE PROBLEM.C + exp subjec o condiion y y y a We recognize is as a linear firs order differenial equaion wi variable coefficiens. We divide bo sides by, assuming 0 (if 0 e derivaive disappears from e equaion) y y + exp Muliply bo sides by a facor, presumed posiive y+ y exp We wan suc a d ln d ln d c were c is an arbirary consan expcexp ln Le s pic c 0 exp Plugging in e muliplying facor d y d exp exp exp

4 d exp y d y c y c exp wen 0; exp wen 0; We ave iniial condiion y a a exp c exp a c exp y exp a wen 0 y exp a wen 0 exp

5 SOLUTION TO THE PROBLEM A model a describes e populaion of a fisery in wic arvesing aes place a a consan rae is given by P P were and are posiive consans. Noe: we assume a we can ave non-wole number of fis, e.g..5 fis, fis. A. Solve e differenial equaion for P subjec o e condiion P0 P P is a linear firs order ODE wi consan coefficiens P P a. P P We recognize e cain rule, P d ln P P d, and plug is ino e above. d ln P d Inegraing bo sides wi respec o ln P c were c is an arbirary real consan Exponeniaing bo sides we ge: P expcexp In case P 0 we ave P expcexp In case P 0 we ave P expcexp

6 Since exp c 0, we can wrie e general soluion wiou e absolue values as were is an arbirary real consan exp y Plugging in e condiion y 0 a a a. Te soluion saisfying e given condiion is y a exp B. Describe e beavior of e populaion P for increasing ime in e following ree cases: case : a ; case. a ; and case 3. 0 a. CASE : Wen a, e populaion size increases wiou bound via an exponenial grow. CASE : Wen a, e populaion size remains consan a y CASE 3: Wen a, we ave a 0, and e populaion size decreases unil exincion. Acually, if we ad a a 0 we would ave negaive number of fis, wic wouldn mae sense. C. In wic of e ree cases considered above, wic will ever ave e populaion go exinc in a finie ime? In oer words, does ere exiss a ime e suc a 0 P in any of ese cases? If e populaion goes exinc in any of e ree cases, en find e ime e wen i does so. e In e ird case ( a ), ere is a possibiliy of exincion. Nex we derive e formula for e ime of exincion e.

7 y a expe 0 a expe a a exp exp exp a e a log e e e Of course, afer exincion e populaion canno grow anymore and us i only maes sense for P P e populaion governing equaion o apply only wen P 0. If we applied wen P 0, en we would go ino negaive populaion size, wic would be non-sense.

8 SOLUTION TO THE PROBLEM 3: A sydiver jumps from 00 meers above e ground and falls owards Ear under e influence of graviy. Assume e force due o air resisance is proporional o e velociy of e paracuis, wi differen consans of proporionaliy wen e paracue is closed ( 5 ) and open ( 5 ). Assume a e paracuis couns off en seconds before opening e paracue. We ave e following differenial equaion governing e velociy of e paracuis as a funcion of ime before e paracue opens: mv mg v ; and we ave e following equaion governing e velociy of. Mass of e e paracuis as a funcion of ime wile e paracue is open: mv mg v paracuis is m 88 g, graviaional acceleraion is downward direcion). Te quesions o answer are on e nex page g 9.8 m s (i is negaive because i is in a A. Plo e velociy and disance from e ground as a funcion of ime for e firs 45 seconds of e flig. B. Wa is e velociy wen e paracue is deployed? C. How long does e jump las (esimae from plo)? D. Wa is e velociy a landing (esimae)? Te following conains soluions o all pars (A,B,C,D): Firs we wrie down e differenial equaion governing e fall during e firs 0 seconds (assuming v 0 0) mv mg v m v v mg v mg v m d mg ln v d m

9 mg ln v c m mg v expcexp m mg exp were is an arbirary real number m v We find e value of a saisfies e iniial condiion v 0 0 mg mg 0 mg mg mg v exp exp m m Tis equaion is valid for e firs en seconds, i.e. for 0 0 sec So e velociy a ime e paracue opens is v mg m sec exp 0 exp meers per second Nex we wrie down e differenial equaion governing e fall wile e paracue is open mv mg v Using e same ecnique as above (jus excanging for ) mg exp were is an arbirary real number m v We find e value of a saisfies e condiion v 0 sec v P meers per second

10 v P mg m exp 0 were is an arbirary real number mg vp exp 0 m mg mg v vp exp0 exp olds for 0 m m mg mg v vp exp 0 olds for 0 m Disance ravelled is an inegral of velociy Firs we compue disance ravelled during e firs 0 seconds. landing landing mg m g mg exp exp 0 m m 0 s d m g mg m g m exp exp s olds for e firs en seconds, i.e. for 0 0 sec Nex we compue disance ravelled afer paracue opens.

11 velociy in meers per second mg mg s vp exp 0 d 0 m m mg mg m vp exp 0 0 m mg mg m mg mg m vp exp 0 vp 0 Now we can plo velociy and disance from e ground for e firs 45 seconds (say using Malab) ime in seconds

12 disance from ground in meers ime in seconds To find velociy a landing and ime of flig, we need a plo for a longer inerval.

13 disance from ground in meers ime in seconds