- 4.3.1. Use separation of variables to solve a differential equation.
- 4.3.2. Solve applications using separation of variables.
We now examine a solution technique for finding exact solutions to a class of differential equations known as separable differential equations. These equations are common in a wide variety of disciplines, including physics, chemistry, and engineering. We illustrate a few applications at the end of the section.
Separation of Variables
We start with a definition and some examples.
A separable differential equation is any equation that can be written in the form
The term ‘separable’ refers to the fact that the right-hand side of the equation can be separated into a function of times a function of Examples of separable differential equations include
The second equation is separable with and the third equation is separable with and and the right-hand side of the fourth equation can be factored as so it is separable as well. The third equation is also called an autonomous differential equation because the right-hand side of the equation is a function of alone. If a differential equation is separable, then it is possible to solve the equation using the method of separation of variables.
- Check for any values of that make These correspond to constant solutions.
- Rewrite the differential equation in the form
- Integrate both sides of the equation.
- Solve the resulting equation for if possible.
- If an initial condition exists, substitute the appropriate values for and into the equation and solve for the constant.
Note that Step 4. states “Solve the resulting equation for if possible.” It is not always possible to obtain as an explicit function of Quite often we have to be satisfied with finding as an implicit function of
Using Separation of Variables
Find a general solution to the differential equation using the method of separation of variables.
Use the method of separation of variables to find a general solution to the differential equation
Solving an Initial-Value Problem
Using the method of separation of variables, solve the initial-value problem
Find the solution to the initial-value problem
using the method of separation of variables.
Applications of Separation of Variables
Many interesting problems can be described by separable equations. We illustrate two types of problems: solution concentrations and Newton’s law of cooling.
Consider a tank being filled with a salt solution. We would like to determine the amount of salt present in the tank as a function of time. We can apply the process of separation of variables to solve this problem and similar problems involving solution concentrations.
Determining Salt Concentration over Time
A tank containing of a brine solution initially has of salt dissolved in the solution. At time another brine solution flows into the tank at a rate of This brine solution contains a concentration of of salt. At the same time, a stopcock is opened at the bottom of the tank, allowing the combined solution to flow out at a rate of so that the level of liquid in the tank remains constant (Figure 4.16). Find the amount of salt in the tank as a function of time (measured in minutes), and find the limiting amount of salt in the tank, assuming that the solution in the tank is well mixed at all times.
A tank contains kilograms of salt dissolved in liters of water. A salt solution of is pumped into the tank at a rate of and is drained at the same rate. Solve for the salt concentration at time Assume the tank is well mixed at all times.
Newton’s law of cooling
Newton’s law of cooling states that the rate of change of an object’s temperature is proportional to the difference between its own temperature and the ambient temperature (i.e., the temperature of its surroundings). If we let represent the temperature of an object as a function of time, then represents the rate at which that temperature changes. The temperature of the object’s surroundings can be represented by Then Newton’s law of cooling can be written in the form
The temperature of the object at the beginning of any experiment is the initial value for the initial-value problem. We call this temperature Therefore the initial-value problem that needs to be solved takes the form
where is a constant that needs to be either given or determined in the context of the problem. We use these equations in Example 4.13.
Waiting for a Pizza to Cool
A pizza is removed from the oven after baking thoroughly, and the temperature of the oven is The temperature of the kitchen is and after minutes the temperature of the pizza is We would like to wait until the temperature of the pizza reaches before cutting and serving it (Figure 4.17). How much longer will we have to wait?
A cake is removed from the oven after baking thoroughly, and the temperature of the oven is The temperature of the kitchen is and after minutes the temperature of the cake is
- Write the appropriate initial-value problem to describe this situation.
- Solve the initial-value problem for
- How long will it take until the temperature of the cake is within of room temperature?
Section 4.3 Exercises
Solve the following initial-value problems with the initial condition and graph the solution.
Find the general solution to the differential equation.
Find the solution to the initial-value problem.
For the following problems, use a software program or your calculator to generate the directional fields. Solve explicitly and draw solution curves for several initial conditions. Are there some critical initial conditions that change the behavior of the solution?
Most drugs in the bloodstream decay according to the equation where is the concentration of the drug in the bloodstream. If the half-life of a drug is hours, what fraction of the initial dose remains after hours?
A drug is administered intravenously to a patient at a rate mg/h and is cleared from the body at a rate proportional to the amount of drug still present in the body, Set up and solve the differential equation, assuming there is no drug initially present in the body.
[T] How often should a drug be taken if its dose is mg, it is cleared at a rate mg/h, and mg is required to be in the bloodstream at all times?
A tank contains kilogram of salt dissolved in liters of water. A salt solution of kg salt/L is pumped into the tank at a rate of L/min and is drained at the same rate. Solve for the salt concentration at time Assume the tank is well mixed.
A tank containing kilograms of salt dissolved in liters of water has two salt solutions pumped in. The first solution of kg salt/L is pumped in at a rate of L/min and the second solution of kg salt/L is pumped in at a rate of L/min. The tank drains at L/min. Assume the tank is well mixed. Solve for the salt concentration at time
[T] For the preceding problem, find how much salt is in the tank hour after the process begins.
Torricelli’s law states that for a water tank with a hole in the bottom that has a cross-section of and with a height of water above the bottom of the tank, the rate of change of volume of water flowing from the tank is proportional to the square root of the height of water, according to where is the acceleration due to gravity. Note that Solve the resulting initial-value problem for the height of the water, assuming a tank with a hole of radius ft. The initial height of water is ft.
For the preceding problem, determine how long it takes the tank to drain.
For the following problems, use Newton’s law of cooling.
The liquid base of an ice cream has an initial temperature of before it is placed in a freezer with a constant temperature of After hour, the temperature of the ice-cream base has decreased to Formulate and solve the initial-value problem to determine the temperature of the ice cream.
[T] The liquid base of an ice cream has an initial temperature of before it is placed in a freezer with a constant temperature of After hours, the temperature of the ice-cream base has decreased to At what time will the ice cream be ready to eat? (Assume is the optimal eating temperature.)
[T] You are organizing an ice cream social. The outside temperature is and the ice cream is at After minutes, the ice cream temperature has risen by How much longer can you wait before the ice cream melts at
You have a cup of coffee at temperature and the ambient temperature in the room is Assuming a cooling rate write and solve the differential equation to describe the temperature of the coffee with respect to time.
[T] You have a cup of coffee at temperature that you put outside, where the ambient temperature is After minutes, how much colder is the coffee?
You have a cup of coffee at temperature and you immediately pour in part milk to parts coffee. The milk is initially at temperature Write and solve the differential equation that governs the temperature of this coffee.
You have a cup of coffee at temperature which you let cool minutes before you pour in the same amount of milk at as in the preceding problem. How does the temperature compare to the previous cup after minutes?
Solve the generic problem with initial condition
Prove the basic continual compounded interest equation. Assuming an initial deposit of and an interest rate of set up and solve an equation for continually compounded interest.
Assume an initial nutrient amount of kilograms in a tank with liters. Assume a concentration of kg/L being pumped in at a rate of L/min. The tank is well mixed and is drained at a rate of L/min. Find the equation describing the amount of nutrient in the tank.
Leaves accumulate on the forest floor at a rate of g/cm2/yr and also decompose at a rate of per year. Write a differential equation governing the number of grams of leaf litter per square centimeter of forest floor, assuming at time there is no leaf litter on the ground. Does this amount approach a steady value? What is that value?
Leaves accumulate on the forest floor at a rate of g/cm2/yr. These leaves decompose at a rate of per year. Write a differential equation governing the number of grams of leaf litter per square centimeter of forest floor. Does this amount approach a steady value? What is that value?