Applications of first-order and simple higher-order


Applications of first-order and simple higher-order DEs

Mechanics and Electric circuits

1. A mass of m Kg falls vertically downward under the influence of gravity starting from rest. Assuming that air resistance is negligible, set up a DE. and the associated conditions describing the motion.

2. A ball is thrown vertically upward with an initial velocity of 128 ft/sec. What is its velocity after 2, 4, and 6 sec? When will it return to its starting position? What is the maximum height to which it will rise before returning?

3. A sky diver (and of course her parachute) falls from rest. The combined weight of the sky diver and parachute is W. The parachute has a force acting on it (due to air resistance) which is proportional to the speed at any instant during the fall.

Assuming that she falls vertically downward and that the parachute is already open when the jump takes place, describe the ensuing motion.

4. A generator having emf 100 volts is connected in series with a 10 ohm resistor and an inductor of 2 henries. If the switch K is closed at time t = 0, set up a differential equation for the current.

5. A decaying emf E = 200e

-5t is connected in series with a 20 Ω resistor and 0.01 farad capacitor. Assuming Q = 0 at t = 0, set up the differential equation that will help you find the charge and current at any time. How would you find out that the charge reaches a maximum? How would you calculate it?


6. A tank is filled with 10 gallons of brine in which is dissolved 5 lb of salt. Brine containing 3 lb of salt per gallon enters the tank at 2 gal per minute, and the wellstirred mixture leaves at the same rate.

(a) Find the amount of salt in the tank at any time.

(b) How much salt is present after 10 minutes?

(c) How much salt is present after a long time (i.e., as t →∞ )

7. Two chemicals, A and B, react to form another chemical C. It is found that the rate at which C is formed varies as the product of the instantaneous amounts of chemicals A and B present. The formation requires 2 lb of A for each pound of B.

If 10 lb of A and 20 lb of B are present initially, and if 6 lb of C are formed in 20 minutes, find the amount of chemical C at any time.

The Hanging Cable

8. A flexible cable of small (negligible) weight supports a uniform bridge (see figure below). Determine the shape of the cable.

9. A flexible rope having constant density hangs between two fixed points.

Determine the shape of the rope.

Flight of a Projectile

10. A projectile is fired from a cannon which is inclined at an angle θ


with the horizontal and which imparts to the projectile a muzzle velocity having magnitude v


. Assuming no air resistance and a flat stationary earth, describe the motion of the flight of the projectile and answer the following:

(a) What is the total time of flight of the projectile?

(b) What is the range of the projectile?

(c) What is the maximum height reached?

(d) What type of curve does the projectile describe?


11. According to Newton's universal law of gravitation, any two objects separated by a distance r and having masses M


and M


, respectively, are attracted toward each other with a force having magnitude given by F =

GM r





where G is a universal gravitational constant. Make use of this law to describe the motion of our planet about the sun. Neglect the effects of all other planets.


12. Find the shape of a reflector that reflects all light rays emitted by a point source parallel to a fixed line.

Geometry of a Rotating Liquid

13. A right circular cylinder having vertical axis is filled with water and is rotated about its axis with constant angular velocity ω . What shape does the water surface take?