Question

Please answer all parts in detail, thank you.

Find the orthogonal trajectories of the family of curves y2 = kx5

Answer 1

Solutions - 0 criven en is y² = kas k= 8 Now decivative o dieta. on both side of Egh 2 dy sy - 5kg4 dn dy an Sky4 ) ) 342 3 put k= & In Ear as dy 5 y ln 14 11 зу? Sy dy du 3x for of the genal trajectories, dy Tu dy da 3x dy du sy sy dy = – 3nda Scanned by TapScanner

Integrating get. 5 - 24 & c' c'- constant of integration, مااه 2 - 3 y etc do, y=MP4c is the orthogonal trajectories of the family of curves yžkas gooo solution: - Given that callying capacity Ko gooo The biologists stocked the lake with 300 fish - Plo)- 300. Figh trippled in the first year P (1)= 3x300 a goo size of the population satisfied the legistic Eqh. dp Scanned by TapScanner

d = kp ( 1-2 9000 de kat Plie 9000 Integrating we get dp k sdt to Pli-9/gooo) k fat to S S dp. 9000 - p² gooop qooo de = & Sat &c (900-POP by wring particel Fraction in its 9000 f P (2000-P) P (9000-1) АРҒ B09000- gooo = fos p= quoo 9000 A-9000 to A-1 Scanned by TapScanner

for p=1 9000 = 11+ B (9000-1) (900-1) = B (9000-1) - 1B = 1 . 9ooo = + ! (9000-P) p (9000-P) P dp fc = s de P & Sat - f (9000-P) lap - ln (9000-p) = kt tc. dn Р 9000-P = kffc kt fc P 900-P P Ckt - 9000 p Р 9000-P - се P 9000-P Cokta et p = 9000-p Scanned by TapScanner

Pflekt p = 9000 Plit cett- 9000 quoo Plt) where ItCooke years. Scanned by TapScanner