# Differential Equations : Separable Variables

## Example Questions

### Example Question #1 : First Order Differential Equations

Solve the given differential equation by separation of variables.

Explanation:

To solve this differential equation use separation of variables. This means move all terms containing  to one side of the equation and all terms containing  to the other side.

First, multiply each side by .

Now divide by  on both sides.

Next, divide by  on both sides.

From here take the integral of both sides. Remember rules for logarithmic functions as they will be used in this problem.

### Example Question #1 : Separable Variables

Solve the following differential equation

Explanation:

So this is a separable differential equation. The first step is to move all of the x terms (including dx) to one side, and all of the y terms (including dy) to the other side.

So the differential equation we are given is:

Which rearranged looks like:

At this point, in order to solve for y, we need to take the anti-derivative of both sides:

Which equals:

And since this an anti-derivative with no bounds, we need to include the general constant C

So, solving for y, we raise e to the power of both sides:

which, simplified gives us our answer:

### Example Question #1 : Separable Variables

Solve the following separable differential equation:  with .

Explanation:

The simplest way to solve a separable differential equation is to rewrite  as  and, by an abuse of notation, to "multiply both sides by dt". This yields

.

Next, we get all the y terms with dy and all the t terms with dt and integrate. Thus,

Combining the constants of integration and exponentiating, we have

The plus minus and the  can be combined into another arbitrary constant, yielding .

Plugging in our initial condition, we have

and

### Example Question #1 : Separable Variables

Solve the general solution for the ODE:

where C is an arbitrary constant

where C is an arbitrary constant

where C is an arbitrary constant

where C is an arbitrary constant

where C is an arbitrary constant

Explanation:

First the differential equation can be separated to:

And then integrated simply to:

### Example Question #1 : Differential Equations

Is the following differential equation separable? If So, how does the equation separate?

The differential equation is separable and becomes:

The differential equation is separable and becomes:

The differential equation is separable and becomes:

This differential equation is autonomous and is therefore not separable.

The differential equation is separable and becomes:

Explanation:

Using exponential rules, we note that  becomes . Meaning that the

differential equation is equivalent to:

which by separation of variables is:

### Example Question #1 : Separable Variables

Is the following differential equation separable, if so, how does the equation separate?

The Differential equation is separable and becomes:

The differential equation is not separable.

The Differential equation is separable and becomes:

The Differential equation is separable and becomes:

The differential equation is not separable.

Explanation:

The differential equation  cannot be written as  and is therefore not separable.

### Example Question #7 : Differential Equations

Solve the given differential equation by separation of variables.

Explanation:

To solve this differential equation use separation of variables. This means move all terms containing  to one side of the equation and all terms containing  to the other side.

First, multiply each side by .

Now divide by  on both sides.

Next, divide by  on both sides.

From here take the integral of both sides. Remember rules for logarithmic functions as they will be used in this problem.

### Example Question #1 : Separable Variables

Solve the following initial value problem: .

Explanation:

This is a separable differential equation. The simplest way to solve this is to first rewrite  as  and then by an abuse of notation to "multiply both sides by dt." This yields . Then group all the y terms with dy and integrate, getting us to . Solving for y, we have . Plugging in our condition, we find . Raising both sides to the power of -1/3, we see . Thus, our final solution is

### Example Question #1 : Separable Variables

Solve the following equation

Explanation:

This is a separable ODE, so rearranging

Integrating

Plugging in the initial condition and solving gives us

Solving for  gives us