Differential Equations : Introduction to Differential Equations

Study concepts, example questions & explanations for Differential Equations

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Example Questions

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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

Explanation:

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:

Explanation:

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

Explanation:

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 .  

Explanation:

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

Explanation:

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

Explanation:

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:

Explanation:

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:

Explanation:

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:

Explanation:

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:

Explanation:

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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