Differential Equations Worksheet
 

1) Show that  is a solution to the differential equation, 

a) 
 

Solution:

Separate variables: 
 

Integrate both sides with respect to x: 
 

 

At this point you are actually done. But you can proceed to express y as a function of x as shown below.

a) The rate of decay radioactive nuclei is proportional to the number of nuclei present in a given sample. Using N as the number of nuclei, write the above relationship in the form of a differential equation.

b) Separate the variables, N and t, in the above expression.

c) Determine the solution for the differential equation.
 
 

d) Write a function that expresses the number of nuclei present at any time, t, as a function of the initial number of nuclei, N_o and time.

e) The half-life is the time required for one-half of a sample to decay. Derive an expression for half-life, t_1/2, using the above expression.

f) The charcoal from a tree killed in the volcanic eruption that formed Crater Lake in Oregon contained 44.5% of the carbon-14 found in living matter. About how old is Crater Lake? The half-life of carbon-14 is about 5700 years.

Given the half-life, the proportionality constant can be found:
 

        Note that k is negative since this is a decay process. To find the age, solve for t in the following:

5) Newton's Law of Cooling

Newton's Law of cooling states that the rate of change of temperature T of an object in a medium of a different temperature, T_f, is proportional to the difference between T and T_f.

a) Express the above relationship as a differential equation.

b) Define the difference in temperatures as y and express the rate of change of y in terms of T and T_f.

c) Using the above relationship, write the differential equation that relates the dy/dt to your original proportionality constant and y.

d) Solve the differential equation so that you obtain an equation that gives the difference in temperatures, y, as a function of time.

e) Now write an equation that gives the actual temperature as a function of time.

f) A copper ball is heated to a temperature of 100° C. Then at time t = 0, it is placed in water which is maintained at 30° C. After 3 minutes the temperature of the ball is 70° C. At what time will the ball be 31° C?
 
 
 

First, solve for the proportionality constant using T_o, T_f and T(3).

Note that because temperature is decreasing, k must have a negative value.

Now use this value of k to solve for the time required to make T = 31.

6) Torricelli's Law

Torricelli's Law states that water flowing out of a tank will leave at a rate proportional to the square root of the height of the water above the opening. Assume that a cylinder of water with a radius of 1 meter has a spigot at the bottom and is draining. The height of the column can be designated as x.
 
 

a) Write the above relationship in the form of a differential equation that relates the rate of change in the rate of height of the column of water to the height itself.

b) Solve the differential equation by separating variables and integrating.

c) If the height of the water column at time = 0 is 9 m and the constant of proportionality is -0.1, determine the value of the constant of integration.

d) Write an equation that defines the height of the column as a function of time.

e) At what time does is the tank half full?  At what time is it completely drained?

When the tank is empty, x = 0. Setting the above expression equal to zero yields,

When the tank is half full, x = 4.5.

This yields a quadratic equation with two roots: t = 18 and t = 102. Since the tank is empty at 60, the only logical answer is 18.