### All Differential Equations Resources

## Example Questions

### Example Question #1 : Definitions & Terminology

State the order of the given differential equation and determine if it is linear or nonlinear.

**Possible Answers:**

Fourth ordered, linear

Third ordered, nonlinear

Second ordered, nonlinear

Second ordered, linear

Third ordered, linear

**Correct answer:**

Third ordered, linear

This problem contains two questions that need to be solved for: order of the differential equation and whether it is linear or nonlinear.

To determine the order of the differential equation, look for the highest derivative in the equation.

For this particular function recall that,

therefore the highest derivative is three which makes the equation a third ordered differential equation.

The second part of this problem is to determine if the equation is linear or nonlinear. For a differential equation to be linear two characteristics must hold true:

1. The dependent variable and all its derivatives have a power involving one.

2. The coefficients depend on the independent variable .

Looking at the given function,

it is seen that all the variable and all its derivatives have a power involving one and all the coefficients depend on therefore, this differential equation is linear.

To answer this problem completely, the differential equation is a linear, third ordered equation.

### Example Question #1 : Definitions & Terminology

Which of the following three equations enjoy both local existence and uniqueness of solutions for any initial conditions?

**Possible Answers:**

**Correct answer:**

By the cauchy-peano theorem, for , as long as is continuous on a closed rectangle around our starting point, we have local existence. All three functions are continuous everywhere, so they enjoy local existence at every starting point.

We can show that the solutions to differential equations are unique by showing that is Lipschitz continuous in y. If is continuous, then this will suffice to show the Lipschitz continuity.

Note that the first and third equations are continuous for all y and t, but that the second is not continuous when . More concretely, when , both the equation and the equation would satisfy the differential equation.

### Example Question #3 : Definitions & Terminology

State the order of the given differential equation and determine if it is linear or nonlinear.

**Possible Answers:**

Third ordered, nonlinear

Second ordered, nonlinear

Fourth ordered, linear

Third ordered, linear

Second ordered, linear

**Correct answer:**

Third ordered, linear

This problem contains two questions that need to be solved for: order of the differential equation and whether it is linear or nonlinear.

To determine the order of the differential equation, look for the highest derivative in the equation.

For this particular function recall that,

therefore the highest derivative is three which makes the equation a third ordered differential equation.

The second part of this problem is to determine if the equation is linear or nonlinear. For a differential equation to be linear two characteristics must hold true:

1. The dependent variable and all its derivatives have a power involving one.

2. The coefficients depend on the independent variable .

Looking at the given function,

it is seen that all the variable and all its derivatives have a power involving one and all the coefficients depend on therefore, this differential equation is linear.

To answer this problem completely, the differential equation is a linear, third ordered equation.

### Example Question #1 : Definitions & Terminology

Which of the following definitions describe an autonomous differential equation.

**Possible Answers:**

A differential equation that does not depend explicitly on the independent variable of the equation; usually denoted or .

A differential equation that does not depend explicitly on the dependent variable of the equation; usually denoted .

A differential equation that has Eigen Values of 0.

A differential equation that models growth exponentially.

**Correct answer:**

A differential equation that does not depend explicitly on the independent variable of the equation; usually denoted or .

By definition, an autonomous differential equation does not depend explicitly on the independent variable. An autonomous differential equation will take the form

### Example Question #5 : Definitions & Terminology

State the order of the given differential equation and determine if it is linear or nonlinear.

**Possible Answers:**

Third ordered, linear

Second ordered, nonlinear

Third ordered, nonlinear

Second ordered, linear

Fourth ordered, linear

**Correct answer:**

Third ordered, linear

This problem contains two questions that need to be solved for: order of the differential equation and whether it is linear or nonlinear.

To determine the order of the differential equation, look for the highest derivative in the equation.

For this particular function recall that,

therefore the highest derivative is three which makes the equation a third ordered differential equation.

The second part of this problem is to determine if the equation is linear or nonlinear. For a differential equation to be linear two characteristics must hold true:

1. The dependent variable and all its derivatives have a power involving one.

2. The coefficients depend on the independent variable .

Looking at the given function,

it is seen that all the variable and all its derivatives have a power involving one and all the coefficients depend on therefore, this differential equation is linear.

To answer this problem completely, the differential equation is a linear, third ordered equation.

### Example Question #1 : Definitions & Terminology

Find Order and Linearity of the following differential equation

**Possible Answers:**

Third Order - Linear

Second Order - NonLinear

None of the other answers.

Third Order - NonLinear

Second Order - Linear

**Correct answer:**

Third Order - Linear

This equation is third order since that is the highest order derivative present in the equation.

This is equation in linear because and derivatives appear to the first power only. and do not affect the linearity.

### Example Question #1 : Initial Value Problems

If is some constant and the initial value of the function, is six, determine the equation.

**Possible Answers:**

**Correct answer:**

First identify what is known.

The general function is,

The initial value is six in mathematical terms is,

From here, substitute in the initial values into the function and solve for .

Finally, substitute the value found for into the original equation.

### Example Question #1 : Initial Value Problems

**Possible Answers:**

**Correct answer:**

So this is a separable differential equation, but it is also subject to an initial condition. This means that you have enough information so that there should not be a constant in the final answer.

You start off by getting all of the like terms on their respective sides, and then taking the anti-derivative. Your pre anti-derivative equation will look like:

Then taking the anti-derivative, you include a C value:

Then, using the initial condition given, we can solve for the value of C:

Solving for C, we get

which gives us a final answer of:

### Example Question #1 : Initial Value Problems

Solve the initial value problem for .

**Possible Answers:**

**Correct answer:**

We have so that and . Solving for y,

and

which we can write because is just another arbitrary constant.

Plugging in our initial value, we have leaving us with a final answer of .

Note, this type of equation pops up frequently in the course and is potentially good to just memorize. For , we have

### Example Question #1 : Initial Value Problems

With

**Possible Answers:**

**Correct answer:**

So this is a separable differential equation with a given initial value.

To start off, gather all of the like variables on separate sides.

Then integrate, and make sure to add a constant at the end

To solve for y, take the natural log, ln, of both sides

Be careful not to separate this, a log(a+b) can't be separated.

Plug in the initial condition to get:

So raising e to the power of both sides:

Solving for C:

giving us a final answer of:

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