Consider the differential equation y’(t)=2y(t)-3 and follow the steps below. a)Draw a direction field on the lines y=0,1,2,3 in the (t,y) plane for (1) b)include equilibrium solutions in the direction field and determine the behavior of y as t->infinity c)does this late-time behavior depend on the initial value of y at t=o? if so, describe the dependency.

3t --- 3t a. Verify that y(t) = b. Verify that yo(t) = = is a solution to y '(t) substituting it into the differential equation. (Enter the terms in the order given.) y' (t) = 5t + 1 2t 4t + 2 = Use angle brackets, ( and ), to indicate vectors, and separate the components with commas. E.g. (e', t², 2e²¹). = 1 -2 2 -2 2 1 2 2 1 1 [ -2 is a particular solution to -2 21 1 2 1 2 y(t) + 2 y(t) by [1 4- (4t+ 4+ 5t) 0 2 (14t+2+ 4t) by substituting it into the differential equation. (Enter the terms in the order given.) Use angle brackets, ( and ), to indicate vectors, and separate the components with commas. E.g. (3, 3t², 2t + 1).

Let u(t) == Consider the modified wave equation my" (t) + kenty(t) = 0. aet. Substitute u(t) into the equation and find appropriate value of a and ẞ that transform the differential equation into an equation whose solution is easily determined.

7. Find the functions M(r, y) and (x, y) such that each differential equation is exact: (a) M(r, y)dr + rey + 2ry + dy = 0 1 (b) |dr + N(x, y)dy = 0 2 +y