Question

e critical points for the autonomous equation y'=y (ay) e whether they lead to equilibrium solubions which are shable, unstable or semesbable equilíbria. A solue g =- Jy y (0)=50 X+goo o solve y + y = x 0 Prove that the equation cosy dx - (x sing-y dy=0 s excent and then solve. 2 Prove that the ogrubion (easing) dx+cos ydy=0 is nob excent and then find an anbegrubina factor which will make it exact. Prove that exy dx + 3y dy-0 is an equabion with homogenes Coefficients and solve with an appropriate substitution . Solve the Bernoulli equation y't ax y = bay

Answer 1

2 y = y(2-y) = frag) ferry)=0 y (2- ) = 0 y=0,21 equilibmium for 2 0 fly, co in left of o, and frys 70 in night of a - So x=0 is unstable equilibrium sen for ner flyazo in left of 2, and fuy co so nie in stable equilibrium on? y = -24 yo) = 5. tos vartoh 50 2 --24 7+210 I da 1a+200 ( ay 1-2 da y enlun enly = -2 -2 en la frolt enc enly in (x+ 2001 2 tenc yo CEN+200)-2 yol = 50 50 = c(200)-2 50 (200) 2 Hes Searpred with ly = Px10° LXT 2012 CamScanner

+} = 1 I.F est dx = 6 en son you = frete you = net-etc JE 4-1+ce? Lex, dx – хил.1 – *) 44-Ф. am/sy = -sing N/x = -sing . Diff en is exact. Gen son: I cesy dx + f g 2 dy = c. - 2 wo M cs Scanned with CamScanner