Test 1- first order

You may use your text or other sources, but not other people.  
Show all your work.  Cite all sources.
Don't just state the answer; 
instead, convince me you understand what's going on

For each of problems 1 through 3, the initial condition is
	((Subscript[x, 0]), (Subscript[y, 0])) = (0,0)
and we're interested in at least the region 
	0 ≤ x ≤ 3.
(a) Solve each equation analytically if possible or numerically if not.
(b) Plot the solution.

1. 	(dy/dx)-y+(e^(-x)) = 0

2.	 (y^2)(dy/dx)-x (e^(-(x^2)))=0

3.	 (dy/dx)-(e^(-(x^2)))=0

4.	A rocket is shot straight upwards from the earth's surface.

Using energy conservation, the velocity of the rocket at height y above the surface of the earth is
	(dy/dt)=(Sqrt[((2 G M)/(y + R))])
where 
	y = height above the earth's surface
	t = time, with the launch at t=0.
	G = Newton's gravitational constant, 
	M = the mass of the earth, and 
	R = the radius of the earth.  
Let the initial velocity be (Subscript[v, 0]).
Ignore air friction.  (This may not be a good approximation ... but make it anyway.)

(a) Transform this equation into a dimensionless one.
(b) Solve the new equation any way you like.
(c) Explain the possible behaviors as t -> ∞, and describe when each takes place.
(d) Plot the various possible solutions in the problem's phase space.  
(e) Discuss (without actually doing out the details, unless you want the extra brownie points) how the equation and solution would change if a realistic air friction model were included.