Solve Bernoulli ODE with Integrating Factor x^-4 and n=3: Am I Missing a Step?

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    Bernoulli Ode
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Homework Help Overview

The discussion revolves around solving a Bernoulli ordinary differential equation (ODE) of the form x^2*y' + 2xy = 5y^3, with a focus on the use of an integrating factor x^-4 and the parameter n=3. Participants are examining the steps taken to manipulate the equation and are questioning the algebraic transformations involved in reaching the final form of the solution.

Discussion Character

  • Exploratory, Assumption checking, Mathematical reasoning

Approaches and Questions Raised

  • The original poster attempts to apply an integrating factor and expresses confusion about the final algebraic manipulation leading to the solution. Other participants clarify the relationship between y^(-2) and 1/y^2, while one participant reflects on a separate logistics equation problem, indicating a potential shift in focus.

Discussion Status

The discussion is ongoing, with participants providing clarifications and exploring different aspects of the problems presented. There is no explicit consensus on the original poster's algebraic steps, but some guidance has been offered regarding the interpretation of the equation.

Contextual Notes

The original poster is uncertain about the algebraic steps taken in their solution process, and there is a mention of a separate problem involving a logistics equation that introduces additional complexity and potential confusion regarding the application of parameters.

cue928
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So I have the following Bernoulli ODE:
x^2*y' + 2xy = 5y^3
I use an integrating factor of x^-4, my n value is 3. I am okay until I get to the very end, where I get y^-2 = (2+Cx^5)/x; the book shows y^2 = x/(2+Cx^5). Am I must missing an algebra step or did I make a mistake somewhere within?
 
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I see. I was multiplying by -1/2 and wondering why no sq rt on the right hand side.
 
Let me ask you about a logistics equation problem: I got the equation dp/dt = kp^.5 for the following problem: Time rate of change of a population is proportional to sq root of P. At time t = 0, the population numbers 100 rabbits and is increasing at rate of 20 rabbits per month. How many will there be after one year? I got a k value of 2 but I don't see where to sub in the rate of change of 20 rabbits per month?
 

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