Call Now to Set Up Tutoring:

(888) 888-0446

Calculus AB : Find General and Particular Solutions Using Separation of Variables

Study concepts, example questions & explanations for Calculus AB

Create An Account

All Calculus AB Resources

45 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #1 : Find General And Particular Solutions Using Separation Of Variables

What is separation of variables?

Possible Answers:

Using algebra to rewrite a differential equation so that two different variables are on opposite sides

A fancy wording for factoring

Integrating with two variables

Taking the derivative of two variables

Correct answer:

Using algebra to rewrite a differential equation so that two different variables are on opposite sides

Explanation:

When we are trying to integrate a differential equation, sometimes we have to use a method called separation of variables. This is because we need to integrate with respect to each variable but are unable to do so when they are on the same side. When we are able to get each variable on different sides with their respective differential (i.e. or ), then we can integrate each side with respect to each variable.

Report an Error

Example Question #1 : Find General And Particular Solutions Using Separation Of Variables

When would one need to use separation of variables?

Possible Answers:

When we are solving a function with two different variables at a certain point

When we need to find the derivative of a differential equation

We never actually have to use separation of variables it is just a shortcut

When we need to integrate a differential equation with two different variables

Correct answer:

When we need to integrate a differential equation with two different variables

Explanation:

By using separation of variables we are able to integrate to solve the differential equation.

Report an Error

Example Question #2 : Find General And Particular Solutions Using Separation Of Variables

Use separation of variables to solve the following differential equation: $\frac{dy}{dx} = 2yx$

Possible Answers:

$y = ce^{x^2}$

$\frac{-1}{2y^2} = x^2 + C$

y = ln|x^2| + C

y +C = x^2 + D

Correct answer:

$y = ce^{x^2}$

Explanation:

We must manipulate this differential equation to get each variable and its own side with its differential. Once we do that we must integrate each side accordingly.

$\frac{dy}{dx} = 2yx$

$\frac{dy}{y} = 2xdx$ (divided by and multiplied by )

$\frac{1}{y}dy = 2xdx$ (just rearranging)

$\int\frac{1}{y}dy = 2\int xdx$ (Let a = D-C )

$ln|y| + C= 2\frac{x^2}{2} + D$

ln|y| = x^2 + a

$e^{ln|y|} = e^{x^2 + a}$

$y = e^{x^2}e^a$ ( e^a is just a constant so we rename it )

$y = ce^{x^2}$

Report an Error

Example Question #3 : Find General And Particular Solutions Using Separation Of Variables

Use separation of variable to solve the following differential equation: $\frac{dy}{dx} = \frac{4x}{y}$

Possible Answers:

y = 4x^2 + c

y = 8x^2 + c

$y = \frac{1}{x^2} + c$

$y = \frac{x^2}{2} + c$

Correct answer:

y = 4x^2 + c

Explanation:

We must manipulate this differential equation to get each variable and its own side with its differential. Once we do that we must integrate each side accordingly.

$\frac{dy}{dx} = \frac{4x}{y}$

ydy = 4xdx (multiplied by and multiplied by )

$\int y dy = \int 4x dx$

$\frac{y^2}{2} + C = \frac{4x^2}{2} + D$ (Let D-C = a )

$\frac{y^2}{2} = \frac{4x^2}{2} + a$

y = 4x^2 + 2a ( is just a constant so we will rename it )

y = 4x^2 + c

Report an Error

Example Question #52 : Differential Equations

Use separation of variable to solve the following differential equations: $\frac{dy}{dx} = \frac{(2 - 3x^2)^2}{y}$

Possible Answers:

$y = \frac{18x^2}{5} - 8x^3 + 8x + c$

$y = ln|\frac{18x^2}{5} - 8x^3 + 8x + c|$

$y = \sqrt{\frac{18x^2}{5} - 8x^3 + 8x + c}$

Correct answer:

$y = \sqrt{\frac{18x^2}{5} - 8x^3 + 8x + c}$

Explanation:

We must manipulate this differential equation to get each variable and its own side with its differential. Once we do that we must integrate each side accordingly.

$\frac{dy}{dx} = \frac{(2 - 3x^2)^2}{y}$

y dy = (2-3x^2)^2 dx (multiplied by and )

ydy =9x^4 -12x^2+4 dx (expansion)

$\int y dy= \int 9x^4 - 12x^2 + 4 dx$

$\int y dy= \int 9x^4 dx - \int 12x^2 dx + \int 4 dx$

$\frac{y^2}{2} + C = \frac{9x^5} - 4x^3 + 4x + D$ (Let D-C = a )

$y^2 = \frac{18x^2}{5} - 8x^3 + 8x + 2a$ ( is a constant so call it )

$y = \sqrt{\frac{18x^2}{5} - 8x^3 + 8x + c}$

Report an Error

Example Question #2 : Find General And Particular Solutions Using Separation Of Variables

True or False: We can use separation of variables to solve a differential equation at a particular solution.

Possible Answers:

True

False

Correct answer:

True

Explanation:

Just like integrating for a function of a single variable at a particular solution. We are also able to solve using separation of variables and integrating for our particular solution.

Report an Error

Example Question #6 : Find General And Particular Solutions Using Separation Of Variables

Find the general solution of the following differential equation at the point (1,1) .

$\frac{dy}{dx} = 2x$

Possible Answers:

$y = \sqrt{x}$

y = x^2

y = x^2 + c

y = x

Correct answer:

y = x^2

Explanation:

We first must use separation of variables to solve the general equation, then we will be able to find the general solution.

$\frac{dy}{dx} =2x$

dy = 2 dx (multiplied by )

$\int dy = \int 2x dx$

y + C = x^2 + D (Let D-C = c )

y = x^2 + c

Now we plug in our particular solution (1,1) to solve for our constant

1 = 1^2 + c

0 = c

And so our solution is

y = x^2

Report an Error

Example Question #4 : Find General And Particular Solutions Using Separation Of Variables

Find the general solution of the following differential equation at the point (1,0) .

$\frac{dy}{dx} = \frac{2x^3}{y}$

Possible Answers:

$ln|y| = \sqrt{\frac{x^4}{4} + 1}$

$y = e^{\frac{x^4}{4} + 1}$

$y = \frac{x^4}{4} + 1$

$y = \frac{x^4}{4}$

Correct answer:

$ln|y| = \sqrt{\frac{x^4}{4} + 1}$

Explanation:

$\frac{dy}{dx} = \frac{2x^3}{y}$

ydy = 2x^3dx (multiplying by y^2 and multiplying by )

$\int ydy = 2x^3dx$

$\frac{y^2}{2} + C = \frac{2x^4}{4} + D$ (Let D-C=a )

$\frac{y^2}{2} = \frac{x^4}{2} + a$

$y^2 = \frac{x^4}{4} + \frac{a}{2}$ ( $\frac{a}{2}$ is just a constant so rename it )

$y = \sqrt{\frac{x^4}{4} + c}$

Now we plug in our point (1,0) to solve for .

$1 = \sqrt{\frac{0^4}{4} +c}$

$1 = \sqrt{c}$

1 = c

So our solution at this point is:

$y = \sqrt{\frac{x^4}{4} + 1}$

Report an Error

Example Question #5 : Find General And Particular Solutions Using Separation Of Variables

Find the particular solution for $x ={\pi}$ using the point $(\frac{pi}{2},0)$ of the following differential equation.

$\frac{dy}{dx} = \frac{-sin(x)}{2y}$

Possible Answers:

y = 1

$y = \sqrt{1 + c}$

y =0

$y = \pm 1$

Correct answer:

$y = \pm 1$

Explanation:

We first must use separation of variables to solve the general equation, then we will be able to find the particular solution.

$\frac{dy}{dx} = \frac{-sin(x)}{2y}$

2ydy = -sin(x)dx (multiplying by and )

$\int 2ydy = \int -sin(x)dx$

$\frac{2y^2}{2} + C = cos(x) + D$ (Let D-C = c )

y^2 = cos(x) + c

$y = \sqrt(cos(x) + c)$

Now we plug in our initial condition that we were given $(\frac{pi}{2},0)$

$0 = \sqrt{cos(\frac{\pi}{2}) + c}$

$0 = \sqrt{c}$

0 = c

Now we will solve for when $x = \pi$

$y = \sqrt{cos(x)}$

$y = \sqrt{cos(\pi)}$

$y = \sqrt{1}$

$y = \pm 1$

Report an Error

Example Question #2 : Find General And Particular Solutions Using Separation Of Variables

What do we call a differential equation that can be solved by using separation of variables.

Possible Answers:

separable derivatives

separable partials

separable equations

they have no special name

Correct answer:

separable equations

Explanation:

Separable equations are what we call differential equations that we are able to solve by using separation of variables.

Report an Error

View Tutors

Samir
Certified Tutor

University of South Florida-Main Campus, Master of Fine Arts, Studio Arts.

View Tutors

Grace
Certified Tutor

University of Massachusetts-Dartmouth, Bachelor of Education, Portuguese. University of Massachusetts-Dartmouth, Masters in B...

View Tutors

James
Certified Tutor

St. Mary of the Plains College, Bachelors, Mathematics. Baker University, Masters, Business Administration and Management.

All Calculus AB Resources

45 Practice Tests Question of the Day Flashcards Learn by Concept

Popular Subjects

ISEE Tutors in Seattle, Spanish Tutors in Chicago, Spanish Tutors in Atlanta, SAT Tutors in Miami, ACT Tutors in Denver, Physics Tutors in Chicago, Biology Tutors in Philadelphia, ISEE Tutors in San Francisco-Bay Area, Chemistry Tutors in Philadelphia, Algebra Tutors in Washington DC

Popular Courses & Classes

SSAT Courses & Classes in Seattle, Spanish Courses & Classes in New York City, SSAT Courses & Classes in San Diego, SAT Courses & Classes in Boston, MCAT Courses & Classes in Atlanta, Spanish Courses & Classes in Los Angeles, ACT Courses & Classes in New York City, ACT Courses & Classes in Boston, SSAT Courses & Classes in San Francisco-Bay Area, MCAT Courses & Classes in Seattle

Popular Test Prep

SSAT Test Prep in Boston, ACT Test Prep in Boston, GRE Test Prep in San Francisco-Bay Area, GMAT Test Prep in Dallas Fort Worth, ACT Test Prep in New York City, SAT Test Prep in San Diego, GRE Test Prep in Dallas Fort Worth, GMAT Test Prep in New York City, SAT Test Prep in New York City, GRE Test Prep in Phoenix

DMCA Complaint

If you believe that content available by means of the Website (as defined in our Terms of Service) infringes one or more of your copyrights, please notify us by providing a written notice (“Infringement Notice”) containing the information described below to the designated agent listed below. If Varsity Tutors takes action in response to an Infringement Notice, it will make a good faith attempt to contact the party that made such content available by means of the most recent email address, if any, provided by such party to Varsity Tutors.

Your Infringement Notice may be forwarded to the party that made the content available or to third parties such as ChillingEffects.org.

Please be advised that you will be liable for damages (including costs and attorneys’ fees) if you materially misrepresent that a product or activity is infringing your copyrights. Thus, if you are not sure content located on or linked-to by the Website infringes your copyright, you should consider first contacting an attorney.

Please follow these steps to file a notice:

You must include the following:

A physical or electronic signature of the copyright owner or a person authorized to act on their behalf; An identification of the copyright claimed to have been infringed; A description of the nature and exact location of the content that you claim to infringe your copyright, in \ sufficient detail to permit Varsity Tutors to find and positively identify that content; for example we require a link to the specific question (not just the name of the question) that contains the content and a description of which specific portion of the question – an image, a link, the text, etc – your complaint refers to; Your name, address, telephone number and email address; and A statement by you: (a) that you believe in good faith that the use of the content that you claim to infringe your copyright is not authorized by law, or by the copyright owner or such owner’s agent; (b) that all of the information contained in your Infringement Notice is accurate, and (c) under penalty of perjury, that you are either the copyright owner or a person authorized to act on their behalf.

Send your complaint to our designated agent at:

Charles Cohn Varsity Tutors LLC
101 S. Hanley Rd, Suite 300
St. Louis, MO 63105

Or fill out the form below:

Do not fill in this field

Contact Information

* Full legal name

Company name

* Phone number

* Email address

Address

* Address1

Address2

* City

* State

* Zipcode

* Country

Complaint Details

* URL of copyrighted work (where your original material is located)

* Please describe the copyrighted work so that it may be easily identified

I am the owner, or an agent authorized to act on behalf of the owner of an exclusive right that is allegedly infringed.

I have a good faith belief that the use of the material in the manner complained of is not authorized by the copyright owner, its agent, or the law.

This notification is accurate.

I acknowledge that there may be adverse legal consequences for making false or bad faith allegations of copyright infringement by using this process.

* Signature (your digital signature is legally binding)

HIGH SCHOOL

GRADUATE SCHOOL

K-8

Search 50+ Tests

math tutoring

science tutoring

foreign languages

elementary tutoring

other

Search 350+ Subjects

Calculus AB : Find General and Particular Solutions Using Separation of Variables

Study concepts, example questions & explanations for Calculus AB

All Calculus AB Resources

Example Questions

Example Question #1 : Find General And Particular Solutions Using Separation Of Variables

Example Question #1 : Find General And Particular Solutions Using Separation Of Variables

Example Question #2 : Find General And Particular Solutions Using Separation Of Variables

Example Question #3 : Find General And Particular Solutions Using Separation Of Variables

Example Question #52 : Differential Equations

Example Question #2 : Find General And Particular Solutions Using Separation Of Variables

Example Question #6 : Find General And Particular Solutions Using Separation Of Variables

Example Question #4 : Find General And Particular Solutions Using Separation Of Variables

Example Question #5 : Find General And Particular Solutions Using Separation Of Variables

Example Question #2 : Find General And Particular Solutions Using Separation Of Variables

All Calculus AB Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: