# Solving Separable Differential Equation: dy/dx = (6x^2)/((1+x^3)y)

In summary, the conversation is about finding the value of y in the function dy/dx = (6x^2)/((1+x^3)y), which is a separable function. After integrating, the solution appears to be y^2 = 36ln(1+x^3). However, there is some debate over whether the 36 should be a 4. The conversation also suggests expressing the function in terms of an initial condition.

## Homework Statement

dy/dx = (6x^2)/((1+x^3)y)

## The Attempt at a Solution

it's a separable func. so after integrating, it looks like this (base on my calculation)

y^2 = 36ln(1+x^3)

so how do i find y? is it just sqrt(36ln(1+x^3))

thanks

36? Really? Could you check that again? Aside from that, sure, except y^2=A has two solutions. y=+sqrt(A) and y=-sqrt(A).

Dick said:
36? Really? Could you check that again? Aside from that, sure, except y^2=A has two solutions. y=+sqrt(A) and y=-sqrt(A).

is the 36 supposed to be a 4?

brain fart... suddenly forgot how to do integration..

Well, I got 4. Doesn't mean it's correct. If you're not sure you'd better check again.

I think it's 4 but also I think it's not going to matter too much when you get to the end.

Thanks guys!

i'm pretty sure (lol) it's 4

I suggest you also try writing it in the form x=
and also express both forms in terms of an initial condition e.g. xy=0 and/or yx=0.

## 1. What is a separable differential equation?

A separable differential equation is a type of differential equation where the dependent variable, in this case y, can be written as a product of two functions, each of which depends only on either the independent variable, in this case x, or on y itself.

## 2. How do you solve a separable differential equation?

To solve a separable differential equation, you first need to separate the variables by bringing all x terms to one side and all y terms to the other side. Then, you can integrate both sides with respect to their respective variables and solve for y.

## 3. How do you integrate a separable differential equation?

To integrate a separable differential equation, you can use the basic rules of integration, such as the power rule, product rule, or chain rule. You may also need to use substitution or partial fractions to simplify the integration process.

## 4. Can you provide an example of solving a separable differential equation?

Sure, let's use the given equation: dy/dx = (6x^2)/((1+x^3)y). First, we separate the variables: (1+x^3)y dy = 6x^2 dx. Then, we integrate both sides: ∫ (1+x^3)y dy = ∫ 6x^2 dx. Using the power rule, we get: (y^2)/2 + (x^4)/4 = 2x^3 + C, where C is the constant of integration. Finally, solving for y, we get: y = √(4x^6 + 8x^3 + C).

## 5. What are some real-life applications of separable differential equations?

Separable differential equations are commonly used in various fields of science and engineering, such as physics, chemistry, biology, and economics. They can be used to model growth and decay processes, population dynamics, chemical reactions, and many other natural phenomena.

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