$(15$ points$)$ We are interested in the following linear first$-$order equation:

$y^{\text{'}}=(1-2t)y$

$($a$)$ Using Python, matplotlib.pyplot as plt and the function plt$.$quiver, plot the direction field for this differential equation on the square $[-2,2][-2,2]$ in the $(t,y)-$plane.

$($b$)$ By observing the direction field, make a hypothesis concerning the limits of solutions as $t+-.$

$($c$)$ Using the method of integrating factors, find the general solution to equation $(1).$

$1$

$($d$)$ Also find the particular solution $y(t)$ which satisfies the initial condition,

$y(0)=-1$

$($e$)$ Plot this solution over the interval $-2,2$ on top of the direction field obtained in part $($a$)$ and check that the direction field is visually tangent to the solution curve at all values of $t.$