💬 👋 We’re always here. Join our Discord to connect with other students 24/7, any time, night or day.Join Here!

Like

Report

Get the answer to your homework problem.

Try Numerade Free for 7 Days

Solve the differential equation.$$y^{2} \frac{d y}{d x}=3 x^{2} y^{3}-6 x^{2}$$

$$\frac{1}{3} \ln \left|y^{3}-2\right|=x^{3}+C$$

Calculus 1 / AB

Calculus 2 / BC

Chapter 7

Integrals and Transcendental Functions

Section 2

Exponential Change and Separable Differential Equations

Functions

Trig Integrals

Campbell University

University of Nottingham

Boston College

Lectures

01:11

In mathematics, integratio…

06:55

In grammar, determiners ar…

08:08

Solve the given differenti…

05:06

03:12

02:11

Solve the differential equ…

03:58

03:46

07:55

04:21

01:10

04:29

to solve this differential equation here. So since we're working with different separable differential equation, you probably want to start more compliant and divided by something. So I've noticed that if we factor out and X squared and let's also factor out a re so you factor out re X squared. That's going to live us with. Leave us with three or why cute minus two. Now, if we were to divide each side by why cubed minus two, we will end up with Why squared over? Why cute Minus two and I'll have d y here and all Multiply the DX over. We have three X squared DX. So now we have this incontestable equation which we can integrate each side like so the right hand side, Well, that's gonna be power rules. It's x cubed. Then we divide by the new power so that the recount so and then we just have our constant C now to integrate, this will need to use. It looks like a use of, but something kind of like natural logs integral. So we might make the guest that you is going to be. Why cubed minus twos of nd you was going to be three. Why square d y? And if we go ahead and divide each side by three So those cancel out and then we get y squared D y is just even to d'you over three. So that looks like it works out. So it should be So factor out that 1/3 then one over you, do you? Which integrates, too? The absolute value of the natural attributes. We have 1/3 natural of the opposite value of you. But that is just why cubed minus two. And this is equal to exe cute plus three. Now, we could just leave it here. What I'm going to do, just multiply each side of this equation by three. So I don't have to look at that fraction. Uh, but you could stop there if you want it, But I'm just gonna rewrite this as natural log of the absolute value of y cute. Minus two is equal to x cute for three excuse because we multiplied by 33 X cube Plus now a constant times. Another constant Still, Constance, though this is gonna be see. So actually, it's called those C one and then this one is just gonna be our constancy, right? And we could keep on going and trying to solve for why, But it's going to get to become not a very pretty looking solution, so I would just leave it like this.

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…

Solve the given differential equation.$$y-x \frac{d y}{d x}=3-2 x^{2} \f…

Solve the given differential equation.$$x^{2} \frac{d y}{d x}=y^{2}+3 x …

Solve the given differential equation.$$y^{\prime \prime}-2 y^{\prime}=6…

Solve the differential equation.$$\frac{d y}{d x}=x y+3 x-2 y-6$$

Solve the given differential equation.$$y^{\prime}+2 x y=2 x^{3}$$

Solve the given differential equation.$$2 x\left(y^{\prime}+y^{3} x^{2}\…

Solve the given differential equation.$$(3 x-2 y) \frac{d y}{d x}=3 y$$<…

Solve the given differential equation.$$\frac{d y}{d x}=\frac{2 x(y-1)}{…

Solve the differential equation.$$\frac{d y}{d x}=3 x^{2} e^{-y}$$

Solve the given differential equation.$$y^{\prime}+6 x^{-1} y=3 x^{-1} y…