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

In summary, the conversation discusses a Bernoulli ODE with an integrating factor of x^-4 and a value of n=3. The individual is unsure if they made a mistake in their algebra as the book shows a different solution. They also discuss a logistics equation problem involving a population of rabbits and the rate of change being proportional to the square root of the population. The individual is unsure where to substitute the given rate of change in their solution.
  • #1
cue928
130
0
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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  • #3
I see. I was multiplying by -1/2 and wondering why no sq rt on the right hand side.
 
  • #4
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?
 

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

1. How do I know if I am missing a step when solving the Bernoulli ODE with an integrating factor of x^-4 and n=3?

The best way to ensure that you are not missing a step is to carefully check your work and make sure that all of the steps in the solving process have been completed correctly. You can also double check your answer by plugging it back into the original equation and seeing if it satisfies the equation.

2. What is the purpose of using an integrating factor when solving a Bernoulli ODE?

The purpose of using an integrating factor is to transform the Bernoulli equation into a linear equation, which is easier to solve. It also helps to eliminate any potential singularities in the equation.

3. Can I use a different integrating factor for solving a Bernoulli ODE with a different value of n?

Yes, the integrating factor for a Bernoulli ODE can vary depending on the value of n. It is important to choose an integrating factor that will transform the equation into a linear form.

4. What are the steps for solving a Bernoulli ODE with an integrating factor of x^-4 and n=3?

The steps for solving a Bernoulli ODE with an integrating factor of x^-4 and n=3 are as follows:
1. Identify the equation as a Bernoulli ODE.
2. Rewrite the equation in the form dy/dx + P(x)y = Q(x)y^n.
3. Find the integrating factor by taking the inverse of the coefficient of y^n and raising it to the power of n-1. In this case, the integrating factor will be x^-4.
4. Multiply both sides of the equation by the integrating factor.
5. Simplify the equation and rearrange it into a linear form.
6. Solve the linear equation by using standard methods.
7. Check your solution by plugging it back into the original equation.

5. Can I solve a Bernoulli ODE without using an integrating factor?

It is possible to solve a Bernoulli ODE without using an integrating factor, but it may be more difficult and time consuming. The integrating factor helps to simplify the equation and make it easier to solve, so it is recommended to use one whenever possible.

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