Question

3. (10 points) Solve the following Bernoulli Equation. ty+2y- y = 0 1>0. 4. (10 points) Solve each of the following and tell whether the differential equation is linear or nonlinear. 2 1 y(i)=0

Answer 1

3. 4 given Bernoulli equation is 온3g' +24화 -y to 4장' 424=g9 a' + 남성에 + 해 끝 1 t3) - 군 put the 아구 룺 뼱 dz वर 은 님에 वर - वर From get 21 g 습 1 를 하 허은 국역 4 룸근 t3 वर and +3 = 2 6 - 1 t: = - 유 이모 . ना 1 This Linear equation in z Integrating factor (IF) 아 튿야 T: 14 a differential equations of the m a 6 logt doma oleme + Pctly - logt Then 2 + I.F= 2 Sptat Hee Pt=-월

1. F 3 zot +-6 dt +3. s 2 - 35 - 3640) 3 8 + 8 + c on 2 76 3+6 8 +8 teto - Za it's denivative is of degree - Multiplying both sides of by and integrating get t dt + C, where c's an arbitrary constant 2- 45] ty the given Bennout The as an equation is y (4) =0 y and equations 3 842 acts solution is equation 3 ya 8+2+et6, 4. Where arbitrary constant. given differential y' + 2 + y = the dependent 2t e variable Since so Linear in given differential linear. equation Integrating factor (2.F)

11 sezat 2 logt e e logt² t2 both sides of ay 1. Multiplying Integrating and we get got ² s 14 e 2t. t² at s teetat ezt t. ezt s dt 2 te at 2t 2 2 +C, arbitrary constant. where c is an 2+ yot² = te zt + c y (+) 20 2 2 te = > tezt 2t y. t 2 = 2 e 2 2 y.t² ezt ( 7 2 - ₂4) the given differention solution of equation y y.t² = at e = (1 + 2 - 4 / 4