DE-BOOK Separation of Variables Method (SOV)

Section 4.3 Separation of Variables Method (SOV)

We are now going to discuss the steps involved in the separation of variables method. In order to use this method, the differential equation must be first-order and be separable.

Identifying a differential equation as first-order is as simple as observing the presence of a first derivative and no higher derivatives. In comparison, determining that an equation is separable is not always immediately apparent. Sometimes, it requires some algebra to rearrange the equation into a form where the method can be applied.

Method 2. Separation of Variables (SOV).

If a differential equation can be written in the separable form

\frac{d y}{d x} = f (x) \cdot g (y),

then the general solution can be determined through the steps below.

Step 1: Separate: $\frac{1}{g (y)} d y = f (x) d x$
Step 2: Integrate: $\int \frac{1}{g (y)} d y = \int f (x) d x$
Step 3: Isolate: Isolate $y$ (if possible), and combine constants.

These problems serve as warm-up examples to familiarize you with the separation of variables method. You won’t encounter any tricky algebra or calculus steps, allowing you to focus solely on the general steps needed to apply this technique. However, pay close attention to the new concept of "combining constants", as it will appear frequently throughout this course.

Example 9.

$\frac{d y}{d x} = x^{2}$
Solution. Solution

You actually don’t need separation of variables to solve for $y .$ In fact, you might recognize that finding $y$ is the same as finding the anti-derivative of $x^{2},$ which we know is $y = \frac{1}{3} x^{3} + c .$ Nevertheless, let’s apply separation of variables to make sure we get the same answer.

The differential equation has order 1 and is separable since

$\frac{d y}{d x} = \underset{f (x)}{\underset{↑}{(x^{2})}} \cdot \underset{g (y)}{\underset{↑}{(1)}}$

Step 1: Separate. We use algebra to move all the $y$ terms to the left and all the $x$ terms to the right, giving

$\frac{1}{\underset{g (y)}{\underset{↑}{1}}} \frac{d y}{d x} = \underset{f (x)}{\underset{↑}{x^{2}}}$

Step 2: Integrate. We now integrate both sides with respect to $x,$ like so

$\begin{aligned} \int 1 \underset{= d y}{\underset{⏟}{\frac{d y}{d x} d x}} & = \int x^{2} d x \\ \int 1 d y & = \int x^{2} d x \\ y + c_{1} & = \frac{1}{3} x^{3} + c_{2} . \end{aligned}$

Step 3: Isolate. Finally, we isolate $y$ and combine constants.

$\begin{aligned} y & = \frac{1}{3} x^{3} + c_{2} - c_{1} \end{aligned}$

Since $c_{2} - c_{1}$ is still a constant, we combine it into a single constant, $c,$ and write our solution as

$y = \frac{1}{3} x^{3} + c$
$\frac{d y}{d x} = y^{2}$
Solution. Solution

The differential equation has order 1 and is separable since

$\frac{d y}{d x} = \underset{f (x)}{\underset{↑}{(1)}} \cdot \underset{g (y)}{\underset{↑}{(y^{2})}}$

Step 1: Separate. We use algebra to move all the $y$ terms to the left and all the $x$ terms to the right, giving

$\frac{1}{\underset{g (y)}{\underset{↑}{y^{2}}}} \frac{d y}{d x} = \underset{f (x)}{\underset{↑}{1}}$

Step 2: Integrate. We now integrate both sides with respect to $x,$ like so

$\begin{aligned} \int 1 \frac{1}{y^{2}} \underset{= d y}{\underset{⏟}{\frac{d y}{d x} d x}} & = \int 1 d x \\ \int y^{- 2} d y & = \int 1 d x \\ \frac{y^{- 1}}{- 1} + c_{1} & = x + c_{2} . \end{aligned}$

Step 3: Isolate. Finally, we isolate $y$ and combine constants.

$\begin{aligned} - \frac{1}{y} & = x + c_{2} - c_{1} \\ - \frac{1}{y} & = x - (c_{2} - c_{1}) \\ \frac{1}{y} & = - x + c \leftarrow combined constant \\ y & = \frac{1}{- x + c} \end{aligned}$
$\frac{d y}{d x} + \frac{x}{y^{2}} = 0$
Solution. Solution

$\frac{d y}{d x} = - \frac{x}{y} = - x (\frac{1}{y^{2}}) \leftarrow separable$

Step 1: Separate.

$y^{2} \frac{d y}{d x} = - x$

Step 2: Integrate.

$\begin{aligned} \int y^{2} \underset{= d y}{\underset{⏟}{\frac{d y}{d x} d x}} & = \int - x d x \\ \int y^{2} d y & = \int - x d x \\ \frac{y^{3}}{3} + c_{1} & = - \frac{x^{2}}{2} + c_{2} . \end{aligned}$

Step 3: Isolate.

$\begin{aligned} \frac{y^{3}}{3} & = - \frac{x^{2}}{2} + \underset{= c_{3}}{\underset{⏟}{c_{2} - c_{1}}} \\ y^{3} & = - 3 \cdot \frac{x^{2}}{2} + \underset{= c}{\underset{⏟}{3 \cdot c_{3}}} \\ y^{3} & = - \frac{3}{2} x^{2} + c \\ y & = {(- \frac{3}{2} x^{2} + c)}^{1 / 3} \end{aligned}$

These examples illustrate the fundamental steps of the separation of variables method, giving you a solid foundation for solving simple separable differential equations. As you progress, the techniques you’ve learned here will be essential for tackling more challenging problems. In the next section, we’ll explore more complex examples that require additional algebraic manipulation and deeper understanding.

Reading Questions Check-Point Questions

1. Why is it important to verify if a differential equation is separable?

Why is it important to verify if a differential equation is separable?

Separable equations are easier to solve.
Incorrect. While separable equations can be easier to solve, the main reason to verify separability is to ensure that the separation of variables method can be applied.
It helps to identify the order of the differential equation.
Incorrect. Verifying separability does not necessarily relate to identifying the order of the differential equation.
It simplifies the differential equation.
Incorrect. Verifying separability does not simplify the equation itself but confirms if the equation can be solved by separation of variables.
Not all differential equations are separable, and this method only works on separable equations.
Correct! The separation of variables method is only applicable to separable differential equations, making it important to verify separability first.

2. What are the two key requirements for using the separation of variables method?

What are the two key requirements for using the separation of variables method?

The differential equation must be linear and separable.
Incorrect. Separable is required, but the equation does not need to be linear.
The differential equation must be first-order and linear.
Incorrect. First-order is required, but the equation does not need to be linear for the separation of variables method to be applied.
The differential equation must be second-order and separable.
Incorrect. Separable is required, but the equation cannot be second-order for the separation of variables method to be applied.
The differential equation must be first-order and separable.
Correct! The separation of variables method applies only to first-order differential equations thatare separable.

3. In the differential equation $\frac{d y}{d x} + \frac{x}{y^{2}} = 0,$ which step is crucial for verifying that it is separable?

In the differential equation $\frac{d y}{d x} + \frac{x}{y^{2}} = 0,$ which step is crucial for verifying that it is separable?

Integrating both sides immediately.
Incorrect. Integration should come after the equation is verified to be separable.
Substituting a value for $x .$
Incorrect. Substitution is not part of the separation process; the goal is to separate the variables first.
Expanding the terms on both sides.
Incorrect. Expansion is not necessary here; the focus should be on separating the variables.
Rearranging the equation to isolate $\frac{d y}{d x}$ and separate the variables.
Correct! Rearranging the equation is key to verifying that it can be separated into functions of $x$ and $y .$

4. Separation of Variables Method Steps.

List the order of tasks that you should perform to apply separation of variables method.

Drag each task on the left to the correct order on the right.

Review

Verify the DE is separable.
Task 1
Separate the variables on different sides of the equation.
Task 2
Integrate both sides of the DE
Task 3
Solve for $y,$ if possible
Task 4
Apply initial conditions, if given.
Task 5

You have attempted 1 of 5 activities on this page.

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