We are open Saturday and Sunday!

Call Now to Set Up Tutoring:

(888) 888-0446

Calculus 1 : Solutions to Differential Equations

Study concepts, example questions & explanations for Calculus 1

Create An Account

All Calculus 1 Resources

10 Diagnostic Tests 438 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

← Previous 1 2 … 7 8 9 10 11 12 13 14 15 Next →

Example Question #141 : How To Find Solutions To Differential Equations

Find the derivative of the function y=3x^2+2 .

Possible Answers:

y'(x)= 3x^2

y'(x)= 6x+2

y'(x)= 3x+2

y'(x)= 6x

Correct answer:

y'(x)= 6x

Explanation:

To find the derivative of the function, you must use the power rule

$\frac{\mathrm{d} }{\mathrm{d} x}x^n=nx^{n-1}$ .

Using this rule, the correct answer

$y'=2\cdot 3x^{2-1}$

y'(x)= 6x is obtained.

Report an Error

Example Question #142 : How To Find Solutions To Differential Equations

Find the derivative of the function y=2x^2+5x^4+sin(x) .

Possible Answers:

y'(x)=4x+20x^3-sin(x)

y'(x)=2x+20x^4-cos(x)

y'(x)=4x+20x^3-cos(x)

y'(x)=4x+20x^3+cos(x)

Correct answer:

y'(x)=4x+20x^3+cos(x)

Explanation:

To take the derivative of the first two terms, you use the power rule

$\frac{\mathrm{d} }{\mathrm{d} x}(x^n)=nx^{n-1}$ .

To take the derivative of sin(x), there is the identity that,

sin'(x)=cos(x) .

Using all of the rules and applying,

$y'=2\cdot2x^{2-1}+4\cdot5x^{4-1}+cos(x)$

y'(x)=4x+20x^3+cos(x) .

Report an Error

Example Question #141 : Solutions To Differential Equations

Determine the solution to the following: y'=15x^2y+12xy+y with y(0)=e^4 .

Possible Answers:

$y=\exp(5x^3+6x^2+x)$

The equation is not separable.

$y=\exp(5x^3+6x^2+x+c)$

$y=\exp(5x^3+6x^2+x+4)$

$y=\exp(5x^3+6x^2+x)+4$

Correct answer:

$y=\exp(5x^3+6x^2+x+4)$

Explanation:

Solving a differential equation tends to be quite simple if it is separable. A separable equation is one that can be rewritten in the following form

$\frac{dy}{dx}= f(x)g(y)$ .

From here we can manipulate the equation

$\frac{1}{g(y)}dy=f(x)dx$

and then you can integrate both sides.

We can rewrite the equation as

$\frac{dy}{dx}=(15x^2+12x+1)y$

and we therefore see that it is separable.

Next we manipulate it to get the the y terms on the left hand side and the x terms on the right hand side like the following

$\frac{dy}{y}=(15x^2+12x+1)dx$ .

The next step is to integrate both sides

$\int\frac{dy}{y}=\int(15x^2+12x+1)dx$ .

This gives us

$\ln y +c_1= 5x^3+6x^2+x+c_2$ ,

since the c's are just arbitrary constants we can combine them to get

$\ln y= 5x^3+6x^2+x+c$ .

Since we want to write this as an equation in terms of y we will apply e to both sides.

Therefore we get

$y=\exp(5x^3+6x^2+x+c)$ .

Since the problem gives us an initial condition we solve for c.

Therefore

$\exp(4)=\exp(5\cdot 0^3+6\cdot 0^2 +0+c)$

so c=4.

Thus the solution is

$y=\exp(5x^3+6x^2+x+4)$ .

Report an Error

Example Question #144 : How To Find Solutions To Differential Equations

Solve the following differentiable equation:

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

with a condition of y(1)=0 .

Possible Answers:

y=x^2-x-1

y=x+1

y=x^2

y=x-1

y=x^2-x

Correct answer:

y=x^2-x

Explanation:

When given a first order differential equation to solve, one of the first things to check is that it is homogenous.Homogenous equations are ones in which, given

$\frac{dy}{dx}=f(x,y)$ ,

f(x,y)=f(tx,ty) given any t.

If an equation is homogenous you can do a substitution of y=xz and this new equation will be separable.

The first step is to check if the following equation is homogenous,

$\frac{dy}{dx}= \frac{x+y}{x}$ .

Next we show that the equation is homogenous

$f(x,y)=\frac{x+y}{xy}=\frac{t(x+y)}{tx}=\frac{tx+ty}{tx}=f(tx,ty)$ .

So we now replace y with xz. This gives us the following new equation:

$x\frac{dz}{dx}=\frac{x+xz}{x}= 1+z$ .

This is a separable equation which we manipulate and integrate both sides

$\int\frac{1}{x}dx=\int \frac{1}{z+1}dz$

which leads to

$\ln x = \ln(z+1)+c$ .

Thus we get the equation

x=z+1+c ,

then plugging y back in we get

$x=\frac{y}{x}+1+c$ .

So our family of equations is

y=x^2-x-cx

yet plugging in the given condition gives us that c=0.

Thus the final solution

y=x^2-x .

Report an Error

← Previous 1 2 … 7 8 9 10 11 12 13 14 15 Next →

View Calculus Tutors

Raphael
Certified Tutor

The University of Texas at Dallas, Bachelor of Science, Computer Science.

View Calculus Tutors

Johnathan
Certified Tutor

Central Connecticut State University, Bachelor of Science, Geology. Montana Tech of the University of Montana, Master of Scie...

View Calculus Tutors

Yili
Certified Tutor

Shanghai University of Finance and Economics, Master of Science, Economics.

All Calculus 1 Resources

10 Diagnostic Tests 438 Practice Tests Question of the Day Flashcards Learn by Concept

Popular Subjects

French Tutors in Chicago, Spanish Tutors in Miami, Biology Tutors in Atlanta, Algebra Tutors in Phoenix, GRE Tutors in Denver, French Tutors in Atlanta, GRE Tutors in Dallas Fort Worth, Chemistry Tutors in Washington DC, Chemistry Tutors in Seattle, GRE Tutors in San Francisco-Bay Area

Popular Courses & Classes

GRE Courses & Classes in Boston, SAT Courses & Classes in Washington DC, Spanish Courses & Classes in Dallas Fort Worth, GMAT Courses & Classes in Washington DC, SAT Courses & Classes in San Francisco-Bay Area, Spanish Courses & Classes in Phoenix, GMAT Courses & Classes in Chicago, MCAT Courses & Classes in San Francisco-Bay Area, GRE Courses & Classes in San Diego, Spanish Courses & Classes in Chicago

Popular Test Prep

LSAT Test Prep in Washington DC, MCAT Test Prep in San Francisco-Bay Area, GRE Test Prep in San Diego, ISEE Test Prep in San Diego, SAT Test Prep in Washington DC, SAT Test Prep in Chicago, SAT Test Prep in Miami, GMAT Test Prep in Chicago, MCAT Test Prep in Houston, SSAT Test Prep in San Diego

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 1 : Solutions to Differential Equations

Study concepts, example questions & explanations for Calculus 1

All Calculus 1 Resources

Example Questions

Example Question #141 : How To Find Solutions To Differential Equations

Example Question #142 : How To Find Solutions To Differential Equations

Example Question #141 : Solutions To Differential Equations

Example Question #144 : How To Find Solutions To Differential Equations

All Calculus 1 Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: