What a Separable Differential Equation Is
A separable differential equation is a differential equation that can be rearranged so that all the terms involving are on one side and all the terms involving are on the other.
For example, consider:
This equation is separable because we can rewrite it as:
Now the variables are separated: everything involving is on the left, while everything involving is on the right. Once the variables are separated, we can integrate both sides.
The Basic Strategy
The process is straightforward once you recognize the structure. For an equation of the form , use the following steps:
- Separate the variables. Rearrange the equation so that all terms are on one side and all terms are on the other.
- Integrate both sides. Treat the two sides as separate integrals.
- Include the constant of integration. After integrating, use a single arbitrary constant such as .
- Solve for , if possible.
- Apply the initial condition if one is given to determine the particular solution.
Worked Examples
Solve the differential equation .
Find the particular solution to given .
Solve given the initial condition .
Solve given .
Why the Separation Step Works
The notation can make separation look like ordinary algebra, and in AP Calculus that is usually the most useful way to think about it. More precisely, we are using the relationship between the derivative and differentials to rewrite the equation in an integrable form.
Suppose:
Rearranging gives:
Integrating both sides then produces:
This is the central idea behind separation of variables: turn a differential equation into two ordinary integrals.
Common AP Calculus Patterns
Exponential Growth or Decay
If the rate of change is proportional to the amount present, you often get:
Separating and integrating leads to the familiar exponential model:
Logistic-Type Equations
A common nonlinear model has the form:
These equations can also be separable, although the resulting integral may require partial fractions and is usually more advanced than the basic AP Calculus separation problems.
Equations Involving Powers of
If the right-hand side contains a power of , separation often produces a straightforward power-rule integral. For example:
Common Mistakes
After integrating, you need an arbitrary constant. Write rather than forgetting the constant entirely. If you integrate both sides independently, the two constants can be combined into one.
Before integrating, check that all -dependent factors are on one side and all -dependent factors are on the other. For example, is separated, while is not.
First find the general solution, including the integration constant. Then substitute the initial condition to determine that constant. Skipping the general-solution step often leads to algebra errors.
When , the correct antiderivative is . The absolute value matters unless the problem gives enough information to establish that the quantity inside the logarithm is positive.
Practice Problems
Solve the differential equation .
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Find the particular solution to given .
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Solve given .
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A function satisfies and . Find .
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