Prof. S. Brick Spring ’06 Print your name:
Diﬀerential Equations Exam 1
Math 238 section 101
Show all of your work, and explain your reasoning. 1. Solve y = xy 2 + x
2. Draw the phase diagram of the autonomous diﬀerential equation y = y (5 − y )(y + 2)2 and use it to sketch solution curves. (Here x ≥ 0, but y may be negative.) Identify and classify the types of equilibrium solutions.
3. The vertical motion of an object near the surface of the Earth is subject to two forces: a downward gravitational force FG = −mg (where m is the mass of the object, g is the gravitational constant, and the negative sign represents the downward direction) and a force FR due to air resistance. Assuming that the force due to air resistance is proportional to the velocity and using the fact that the sum of the forces is equal to the product of mass and acceleration, set up a diﬀerential equation for the velocity. Mention the signs of any other constants you use, explaining your reasoning..