Refer a friend and earn $50 when they subscribe to an annual planRefer Now

Get the answer to your homework problem.

Try Numerade Free for 30 Days

Like

Report

For the following problems, find the solution to the initial value problem.$$y^{\prime}=3 y^{2}(x+\cos x), y(0)=-2$$

$y(x)=\frac{-2}{3\left(x^{2}+2 \sin x\right)+1}$

Calculus 2 / BC

Chapter 4

Introduction to Differential Equations

Section 5

First-order Linear Equations

Differential Equations

Baylor University

University of Michigan - Ann Arbor

Boston College

Lectures

01:11

In mathematics, integratio…

06:55

In grammar, determiners ar…

01:10

For the following problems…

03:15

Solve the given initial-va…

01:54

Solving initial value prob…

01:44

Solve the initial-value pr…

04:40

02:58

08:14

06:46

Solve the initial value pr…

01:26

02:10

So we're solving this first order differential equation. This is a nonlinear differential equation, but because I have a wide squared term, but it is a separable equation. So I can write this as, um, we can go ahead and write it as, um why prime over three y squared, they go to x plus co sign of X. So another way of thinking of this is de y the X 1/3. Why squared is equal to X plus co sign of X. So you have one over. Why squared do you want is equal to X plus co sign of X, the X Now it's separated the variables and we can integrate on both sides of this equation. So when you integrate on the left side, you end up with minus one third and you have why, to the negative one. So minus 1/3 why is equal to And then you end up with X squared over two and you integrate the coastline function, you obtain the sine function. So plus a constant of integration. So now to solve this guy for why, um you're going to have Aiken if I multiplied by negative three have won over. Why is equipped in negative? Three X squared over two minus three. Sign of X plus, some constant of integration. Still gotta figure out what that constant is in a moment. So to solve. For why? Why is equal to the reciprocal of all of that? So one over. Negative three X squared over two minus three. Sign of X plus a constant. Now, the boundary condition. The initial condition was wives. Zero is negative. Two. So why zero equal? Negative to. So this means that negative too is equal to one. And, uh, negative. Three times zero over two, minus three times zero plus my constant. So this tells me that, um, So what we see here is negative two. You gotta 1/0 plus zero plus C. Negative two is one. Oversee, therefore C is equal to negative 1/2. So this solution is Why equals one over Negative three X squared over two minus three. Sign of X minus 1/2. Um, you could if you don't like the fractions, they're making it a complex fraction. You could write that a little bit differently. Um, we could do is to say Okay. Well, we could, Right? This is why this equal to let's factor out a negative three and no one ever negative three. And then you got what? X squared over two minus sign of X minus 1/2 and then what I could do it said, Well, I could also multiplied by 2/2. And so then you would get why is equal to one over negative three x squared. What? You would have a two here, um, minus to sign of X minus one and correction there once a factor that that mine is three. This was a positive sign. Um, that you had this, um And then now you see, you got negatives in front of all of those terms there, so you could just write. This is negative to over three x squared plus two sine X plus one. Um, like there. So that would be a final answer. So we had a good final answer when we were right here. It's just that if you want to play with it to not have mixed fractions in the denominator, that fraction, um, made a couple of errors on the way they got to the right answer here. Finally, by factoring out the three and then multiplying by a common denominator

View More Answers From This Book

Find Another Textbook

Numerade Educator

In mathematics, integration is one of the two main operations in calculus, w…

In grammar, determiners are a class of words that are used in front of nouns…

For the following problems, find the solution to the initial value problem.<…

Solve the given initial-value problem.$$y^{\prime \prime}=\cos x, \quad …

Solving initial value problems Find the solution of the following initial va…

Solve the initial-value problem by separation of variables.$$y^{\pri…

Solve the given initial-value problem.$$2 x^{2} y^{\prime}+4 x y=3 \sin …

Solve the given initial-value problem.$$\left(y e^{x y}+\cos x\right) d …

Solve the given initial-value problem:$$y^{\prime \prime}+9 y=5 \cos 2 x…

Solve the initial value problems.$$y^{(4)}=-\cos x+8 \sin 2 x;$$$$y^…

Solve the initial-value problems.$$\frac{d y}{d x}=2+\sin 3 x, y(\pi…

Solving initial value problems Determine whether the following equations are…

03:21

Use the ratio test to determine whether $\sum_{n=1}^{\infty} a_{n}$ converge…

01:32

In the following exercises, use partial fractions to find the power series o…

10:56

In the following exercises, find the radius of convergence of each series,

02:46

In the following exercises, state whether each statement is true, or give an…

01:21

State whether each of the following series converges absolutely, conditional…

02:45

01:25

For the following exercises, evaluate the definite integrals. Express answer…

02:41

For the following exercises, evaluate the integral using the specified metho…

Use the comparison test to determine whether the following series converge. …

04:17

What are the integrating factors for the following differential equations?

92% of Numerade students report better grades.

Try Numerade Free for 30 Days. You can cancel at any time.

Annual

0.00/mo 0.00/mo

Billed annually at 0.00/yr after free trial

Monthly

0.00/mo

Billed monthly at 0.00/mo after free trial

Earn better grades with our study tools:

Textbooks

Video lessons matched directly to the problems in your textbooks.

Ask a Question

Can't find a question? Ask our 30,000+ educators for help.

Courses

Watch full-length courses, covering key principles and concepts.

AI Tutor

Receive weekly guidance from the world’s first A.I. Tutor, Ace.

30 day free trial, then pay 0.00/month

30 day free trial, then pay 0.00/year

You can cancel anytime

OR PAY WITH

Your subscription has started!

The number 2 is also the smallest & first prime number (since every other even number is divisible by two).

If you write pi (to the first two decimal places of 3.14) backwards, in big, block letters it actually reads "PIE".

Receive weekly guidance from the world's first A.I. Tutor, Ace.

Mount Everest weighs an estimated 357 trillion pounds

Snapshot a problem with the Numerade app, and we'll give you the video solution.

A cheetah can run up to 76 miles per hour, and can go from 0 to 60 miles per hour in less than three seconds.

Back in a jiffy? You'd better be fast! A "jiffy" is an actual length of time, equal to about 1/100th of a second.