2 DUE TO FRIDAY OCT 4 AT 10:00AM Problem 2. Let t E R, and consider the function
y(t) = arccos(t) + 41:2.
(a) Find a first order differential equation which has y(t) as a solution, (b) Find a second order differential equation which has y(t) as a solution. Problem 3. (20 pts] Consider the following differential equation: Mt) — 3L“) = (1 +t)4, t > 0. t + 1
(a) Find all the solutions to the differential equation above. (b) Find all the solutions which satisfy y(0) = 3.
(c) Are there any solutions y(t) such that y(0) = 3 and y(1) = 100 ? (d) Are there any solutions y(t) such that y(0) = 3 and y(1) = 36 ? Problem 4. (20 pts) Consider the following differential equation: y'(t) + 2ty(t) = t, t > 0‘
(a) Find all the solutions to the diEerential equation above.
(b) Let yﬁt) be the unique solution such that yl (2) = 0.5, yam the unique solution such that y(0) = 1, and y3(t) the unique solution such that y(0) = 2. Compute
the long term behaviours Hm mt), £13310 ya“), £1323 as) t—l'DO for these three solutions.
(c) Plot the graph of the functions y1(t), yam and y3(t). (d) Is there any solution y(t) which tends to 00 in its long term behaviour ? Problem 5. (20 pts) Consider the following differential equation: t—3
y’ftl292‘?a l>0- (a) Find inﬁnitely many the solutions to the differential equation above. (b) Solve the initial value problem given by the above differential equation and the
initial condition y(1) = —1. (c) Is the constant function y(t) E 0 a solution to the above differential equation ?