First Order Diffy Q Problem with Bernoulli/Integrating Factors

In summary: It looks like the second term is just the antiderivative of the first term, but it's not clear what it should be.
  • #1
The Head
144
2
Homework Statement
If dy/dy + y = (1-1/x)(1/(2y)), solve when x=1, y=1/sqrt(2). Hint: Use a substitution to get into the form of a first order differential equation
Relevant Equations
integrating factor = e^integral(P(x))
v=y^(1-n)
dy/dx +P(x)y=Q(x)y^n
I seem to be getting an unsolvable integral here (integral calculator says it's an Ei function, which I've never seen). My thought was to use Bernoulli to make it linear and then integrating factors. Is that wrong? The basic idea is below:

P(x) 1, Q(x) = 1/2(1-1/x), n=-1, so use v=y^1- -1)=y^2

so y=sqrt(v), dy/dx=1/(2sqrt(v))dv/dx

Thus the equation becomes 1/(2sqrt(v))dv/dx + sqrt(v) = 1/sqrt(v) * 1/2(1-1/x))
Multiplying by 2* sqrt(v) gives dv/dx +2 v = (1-1/x)
Use the integrating factor e^2x gives: d/dx(v*e^(2x)) =e^(2x)(1-1/x) = e^(2x) - e^(2x)*1/x
ve^(2x) then just equals the antiderivative of e^(2x) - e^(2x)*1/x, but that second term seems to give a problem. Is my method incorrect? Not sure how to proceed.
 
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  • #2
I can't spot any mistakes in your solution (except that you got to learn using ##\LaTeX## )so it probably is correct.

I tried wolfram and it also gives the solution in terms of the Ei(2x) function
https://www.wolframalpha.com/input/?i=y'+y=(1-1/x)0.5(1/y)

Maybe check again the exact statement of the problem. There might be some typo after all.
 
  • #3
Assuming you mean ##\frac{dy}{dx}+y=(1-\frac 1x)(\frac 1{2y})## (you wrote dy/dy), it already is a first order ODE.
This does suggest a typo,
 
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  • #4
haruspex said:
Assuming you mean ##\frac{dy}{dx}+y=(1-\frac 1x)(\frac 1{2y})## (you wrote dy/dy), it already is a first order ODE.
This does suggest a typo,
I think there he wanted to write first order linear
 
  • #5
Delta2 said:
I think there he wanted to write first order linear
Yes, that fits. I agree it leads to a nasty integral; I had it in the form of a double exponential.
 

FAQ: First Order Diffy Q Problem with Bernoulli/Integrating Factors

1. What is a First Order Diffy Q Problem with Bernoulli/Integrating Factors?

A First Order Diffy Q Problem with Bernoulli/Integrating Factors is a type of differential equation that involves a first derivative and can be solved using the Bernoulli method or by using integrating factors. It is commonly used in physics and engineering to model real-life situations.

2. What is the Bernoulli method?

The Bernoulli method is a technique for solving first order differential equations of the form dy/dx + P(x)y = Q(x)y^n, where n is any real number except 0 and 1. It involves using a substitution and then solving the resulting linear equation.

3. How do you use integrating factors to solve a First Order Diffy Q Problem with Bernoulli/Integrating Factors?

To use integrating factors, you first need to identify the differential equation as a Bernoulli equation. Then, you multiply both sides of the equation by an integrating factor, which is a function of x that helps to simplify the equation. This will transform the equation into a linear equation, which can then be solved using standard techniques.

4. Can a First Order Diffy Q Problem with Bernoulli/Integrating Factors have multiple solutions?

Yes, a First Order Diffy Q Problem with Bernoulli/Integrating Factors can have multiple solutions. This is because the Bernoulli method involves using a substitution, and different substitutions can lead to different solutions. It is important to check for extraneous solutions when solving these types of equations.

5. What are some real-life applications of First Order Diffy Q Problems with Bernoulli/Integrating Factors?

First Order Diffy Q Problems with Bernoulli/Integrating Factors are commonly used in physics and engineering to model a variety of real-life situations. For example, they can be used to model population growth, radioactive decay, and chemical reactions. They are also used in economics to model supply and demand and in biology to model predator-prey relationships.

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