2 DUE TO FRIDAY OCT 18 AT 10:00AM Problem 2. Consider the following differential equation: y"(t) : 51/“)-
(a) Find all solutions y(t) to the differential equation above. (b) Find two distinct solutions y1(t) and y2(t) such that y1(0) = 0 and 99(0) = 0. (c) How many solutions are there such that y(0) = 0 and y’(0) = 1 ? Problem 3. (20 pts) Consider the following second—order differential equation: 19%) + 33W) = 431(33- (a) Suppose that y(t) = e"15 is a solution to the differential equation above. Find
all possible values of A E C. (b) Find two linearly independent solutions of the differential equation above. In
this context, two solutions y1(t), y2(t) are linearly independent if y1(t) is not a
constant multiple of y2(t). (c) Use superposition to ﬁnd all solutions to the differential equation. (d) Find the unique solution y1(t) such that y1(0) = 0 and yi(0) = 5. (e) Describe the long-term behavior of all solutions to the differential equation.
Problem 4. (20 pts) Consider the following differential equation: y”(t) — 6y’(t) + 10W) = 0. (a) Find the characteristic equation associated to the differential equation above.
(b) Find all solutions to the differential equation above. (0) Is there any constant solution yﬁt) ‘? (d) Find the long-term behaviour Jim W) t—l-DO for all solutions y(t) to the differential equation. (e) Find the unique solution y2(t) satisfying y2(0) : 2 and y§(0) : 9 and plot its
graph. How many zeroes does it have ?