Some old problems possibly repeated

• Find all of the solutions of the differential equation
  $$y'(x)=\left(y(x)\right)^3 e^{-x}.$$

• Find the solution to the Initial Value Problem
  $$2 y'(x) = y(x)+ e^{-x}-2, y(0)=1.$$

• Find the solution to the Initial Value Problem
  $$y''- y ' - 2 y =0, y(0)=0, y'(0)=1.$$

• Suppose $y(t)$ satisfies the Initial Value Problem
  $$y''-2 y + 2 y = 0, y(0)=-1, y'(0)=0.$$
  Without solving for $y(t)$ determine
  • $y''(0)$,
  • $y'''(0)$,
  • $y''''(0)$,
  • $y'''''(0)$.