8.3: Separable Equations
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- Use separation of variables to solve a differential equation.
- 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 Equation
Equation
- 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
Find a general solution to the differential equation
Solution
Follow the five-step method of separation of variables.
1. In this example,
2. Rewrite the differential equation in the form
3. Integrate both sides of the equation:
Let
4. To solve this equation for
Now we use some logic in dealing with the constant
Now exponentiate both sides of the equation (i.e., make each side of the equation the exponent for the base
Again define a new constant
Because of the absolute value on the left side of the equation, this corresponds to two separate equations:
and
The solution to either equation can be written in the form
Since
Note that in writing a single general solution in this way, we are also allowing
5. No initial condition is imposed, so we are finished.
Use the method of separation of variables to find a general solution to the differential equation
- Hint
-
First factor the right-hand side of the equation by grouping, then use the five-step strategy of separation of variables.
- Answer
-
Using the method of separation of variables, solve the initial-value problem
Solution
Follow the five-step method of separation of variables.
1. In this example,
2. Divide both sides of the equation by
3. Next integrate both sides:
To evaluate the left-hand side, use the method of partial fraction decomposition. This leads to the identity
Then Equation
Multiplying both sides of this equation by
4. It is possible to solve this equation for
Next we can remove the absolute value and let a new constant
Then multiply both sides by
Now collect all terms involving
5. To determine the value of
Therefore the solution to the initial-value problem is
A graph of this solution appears in Figure
![A graph of the solution over [-5, 3] for x and [-3, 2] for y. It begins as a horizontal line at y = -2 from x = -5 to just before -3, almost immediately steps up to y = 2 from just after x = -3 to just before x = 0, and almost immediately steps back down to y = -2 just after x = 0 to x = 3.](https://math.libretexts.org/@api/deki/files/7854/imageedit_2_3087334010.png?revision=1)
Find the solution to the initial-value problem
with
- Hint
-
Follow the steps for separation of variables to solve the initial-value problem.
- Answer
-
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.
Solution concentrations
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.
A tank containing

Solution
First we define a function
The general setup for the differential equation we will solve is of the form
INFLOW RATE represents the rate at which salt enters the tank, and OUTFLOW RATE represents the rate at which salt leaves the tank. Because solution enters the tank at a rate of
To calculate the rate at which salt leaves the tank, we need the concentration of salt in the tank at any point in time. Since the actual amount of salt varies over time, so does the concentration of salt. However, the volume of the solution remains fixed at 100 liters. The number of kilograms of salt in the tank at time
The differential equation is a separable equation, so we can apply the five-step strategy for solution.
Step 1. Setting
Step 2. Rewrite the equation as
Then multiply both sides by
Step 3. Integrate both sides:
Step 4. Solve for
Eliminate the absolute value by allowing the constant to be positive, negative, or zero, i.e.,
Finally, solve for
Step 5. Solve for
The solution to the initial value problem is
Note that this was the constant solution to the differential equation. If the initial amount of salt in the tank is
A tank contains
- Hint
-
Follow the steps in Example
and determine an expression for INFLOW and OUTFLOW. Formulate an initial-value problem, and then solve it.Initial value problem:
- Answer
-
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
or simply
The temperature of the object at the beginning of any experiment is the initial value for the initial-value problem. We call this temperature
with
A pizza is removed from the oven after baking thoroughly, and the temperature of the oven is

Solution
The ambient temperature (surrounding temperature) is
with
To solve the differential equation, we use the five-step technique for solving separable equations.
1. Setting the right-hand side equal to zero gives
2. Rewrite the differential equation by multiplying both sides by
3. Integrate both sides:
4. Solve for
5. Solve for
Therefore the solution to the initial-value problem is
To determine the value of
So now we have
Therefore we need to wait an additional
A cake is removed from the oven after baking thoroughly, and the temperature of the oven 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?
- Hint
-
Determine the values of
and then use Equation . - Answer a
-
Initial-value problem
- Answer b
-
- Answer c
-
Approximately
minutes.
Key Concepts
- A separable differential equation is any equation that can be written in the form
- The method of separation of variables is used to find the general solution to a separable differential equation.
Key Equations
- Separable differential equation
- Solution concentration
- Newton’s law of cooling
Glossary
- autonomous differential equation
- an equation in which the right-hand side is a function of
alone
- separable differential equation
- any equation that can be written in the form
- separation of variables
- a method used to solve a separable differential equation