Question

Question 2: Differential Equations a) (3 points) Find the general solution to the equation. Use C,C1,C2 ... to denote arbitrary constants as necessary. y"(t) = sin6t + 20e b) (5 points) Solve the following separable differential equation for the given initial condition. y')= (1) = 0 c) (5 points) Solve the following first-order linear differential equation for the given initial condition. y't) + 7y - 3,y(0) - 1 d) (2 points) State the equilibrium solution and whether it is stable or unstable for part c above.

Answer 1

geleen y (+) = sin 6t + 2044t Aim Find the general solution of D.E() solution from eqn 0 dyllt) sinst +20e4t dt s dy'lt) = S(sinot +20 e 20e4t ]dt y'lt) = -60867 +20e4t+a < (0867 +5e4t 6 4 dy dt tch dy Integrate ($C0867 + Co867 +5e4t+1) dt S dy = 564 00564 +5e4t + ) --- yet) = -7 -t sinet 6 6 +5ett + Gt+Cg 4 sinet yet)= +5 at 5 1 ett tat +Cg 36 4 where I, Cg = are constang Ang b In 3 (t) y' lt) = yel) = 0 tes 2 solution from ean dy dt (In (+)) 3 tel

du (In (t)) 3 il dt ey E Integrate sē ydy = 5 (Init))3 dt t .: Inh) = 2 tot zdz - ey (in (+)4 + 4 → é 4 - (int24 Ano © ylt) + 7 9 = 3 with 960)=1 geen differential Equation is Linear fiost order 0-E , comparing with y' + pit) y = Qit) Plt)= 7 espit)dt 617dt - ezt thee haw 0 (+2= 3 Then PF e espit)dt RF = ett Then general solution Etc P-Fx yet) falt).P.Fdt ezt Jit) = 13e7totta yet)= Şett +4 et ylt)= using yo)=1, we have e yco) tch 3 že + 4 = 2 → 41+) = +(3+4677) to, 19=4 Then 7 7 eft yet) 3 ㅋ At+ 4 7