Finding General Solutions to Separable Differential Equations

What a General Solution Is

A differential equation contains a derivative such as rather than simply giving you a formula for . Solving the differential equation means finding a function that makes the equation true.

When a differential equation has no initial condition, we usually want a general solution: a family of functions containing every solution that can be obtained from the equation. Because integration introduces an arbitrary constant, general solutions normally contain a constant such as .

For AP Calculus, one of the most important types of differential equations that can be solved directly is a separable differential equation.

Recognizing a Separable Differential Equation

A differential equation is separable when you can rewrite it so that all expressions involving are on one side and all expressions involving are on the other.

Once the equation has this form, you can integrate both sides. The key idea is that separation turns a differential equation into an integration problem.

The Basic Method

Separate the variables. Rearrange the differential equation so that the terms are with and the terms are with .
Integrate both sides. Include an arbitrary constant of integration.
Solve for if the problem asks for an explicit general solution.
Check for lost solutions. If you divided by an expression involving , make sure you did not accidentally remove a constant solution.

Why Separation Works

Suppose we have

We can think of the derivative as a ratio of differentials and rearrange the equation:

Now the two variables are separated. Integrating gives:

The constant represents the arbitrary constant that appears when we integrate. This is what creates a family of solutions rather than one particular solution.

Worked Examples

Example 1: A Simple Exponential Equation

Find the general solution to .

Step 1 — Separate: Divide by and multiply by :

Step 2 — Integrate:

Step 3 — Solve for : Exponentiate both sides:

Result:

Example 2: A Polynomial Differential Equation

Find the general solution to .

Step 1 — Separate:

Step 2 — Integrate:

Step 3 — Simplify: Multiply by 2:

Result:

Example 3: A Trigonometric Differential Equation

Find the general solution to .

Step 1 — Separate: Divide by :

Step 2 — Integrate:

Step 3 — Solve for : Since , we can write

Result:

Example 4: A Logistic-Type Differential Equation

Consider the separable differential equation . Find its general solution.

Step 1 — Separate: Divide by and multiply by :

Step 2 — Integrate: Using partial fractions,

Step 3 — Combine logarithms:

Step 4 — Exponentiate:

Step 5 — Solve for :

Result:

General Solution vs. Particular Solution

A general solution contains an arbitrary constant. For example:

Different values of produce different solution curves.

If we are given an initial condition, such as , we can use it to determine the value of the constant. This produces a particular solution.

Example 5: Using an Initial Condition

Suppose the general solution is and the initial condition is .

Step 1: Substitute and :

Step 2: Solve for :

Result: The particular solution is

Be Careful When Dividing

One of the most important details in separable differential equations is that algebraic manipulation can sometimes eliminate a solution.

For example, consider:

If we divide by , we are assuming that . But is itself a constant solution of the original differential equation.

Important AP Calculus Check

Whenever you divide by an expression involving , ask whether that expression could equal zero. If it can, check whether the corresponding value gives a solution that should be included separately.

Common Mistakes

Forgetting the constant of integration

A general solution must normally contain an arbitrary constant. Writing only the antiderivative without gives an incomplete family of solutions.

Putting the variables on the wrong sides

Before integrating, make sure all the -dependent terms are on one side and all the -dependent terms are on the other. The point of separation is to create two integrals, one in each variable.

Forgetting that logarithms need absolute values

When integrating , the general antiderivative is , not simply . Absolute values matter unless the domain is already known to make the argument positive.

Losing a solution when dividing

If you divide by something such as , check the value that makes that expression zero before declaring the solution complete.

Practice Problems

Question 1

Find the general solution to .

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Question 2

Find the general solution to .

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Question 3

Find the general solution to .

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Question 4 (Check for Lost Solutions)

Find all solutions to .

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