Cannot solve - dy/dx = (e^(1/x))/(x*((x+1)^2))

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In summary, the conversation revolves around solving a differential equation involving the function y' = (e^(1/x)) / (x*((x+1)^2)). The individual seeking help has tried multiple methods and resources but has been unable to find a proper solution. Two responders suggest using a u-substitution and integrating the right-hand side of the equation. After some back and forth, the individual is able to successfully solve the equation and shares the solution, which is y = [(x*exp(1/x)) / (x+1)] + k.
  • #1

Homework Statement



Solve the following differential equation:

y' = (e^(1/x)) / (x*((x+1)^2))

Homework Equations



?

The Attempt at a Solution



I wasted over 5 hours trying to solve this equation, but was unable to get a proper solution. Wolfram Alpha and Maple gave me the correct answer (the solution works in the differential equation), but were unable to elaborate. Any help would be appreciated!
 
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  • #2
Welcome to PF.

So you just need to integrate your right hand side. From the looks of it, you should first let [itex] u = \frac{1}{x} [/itex] .
 
  • #3
1. Your ODE is

[tex] \frac{dy}{dx} = \frac{e^{\frac{1}{x}}}{x(x+1)^2} [/tex]

Change the variable [itex] x=\frac{1}{p} [/itex]

Separate the new variables and integrate. What do you get ? Post your work.
 
  • #4
First of all, thank you for the quick replies! I tried a u-substitution again but don't seem to get anywhere:

u = 1/x
dx = - (x^2) du

and v = - (e^u) / (u*((u^-1) + 1)^2) du

And that's as far as I get. Is there something that I am missing?
 
  • #7
I finally got it; this was one insane integral!

From where I left, the denominator had to be simplified and I had to do another integral (by parts) before I got the answer.

The solution was: y = [(x*exp(1/x)) / (x+1)] + k

Thanks for your help Gib Z and bigubau!
 

1. What is the equation dy/dx = (e^(1/x))/(x*((x+1)^2)) and what does it represent?

The equation dy/dx = (e^(1/x))/(x*((x+1)^2)) is a differential equation that represents the rate of change of y with respect to x. It is also known as a first-order separable differential equation.

2. What does it mean when a differential equation cannot be solved?

When a differential equation cannot be solved, it means that a closed-form solution cannot be obtained using standard mathematical methods. This can happen when the equation is too complex or when there is not enough information given.

3. Can this differential equation be solved using numerical methods?

Yes, this differential equation can be solved using numerical methods such as Euler's method or Runge-Kutta methods. These methods involve approximating the solution by using small intervals and calculating the values of y at each interval.

4. What are some applications of this type of differential equation?

This type of differential equation can be used to model various natural phenomena such as population growth, chemical reactions, and radioactive decay. It is also commonly used in physics and engineering to describe the behavior of systems.

5. Are there any real-life situations where this specific equation is applicable?

Yes, this equation can be applied to model the growth of bacteria in a culture, where the rate of growth is dependent on the size of the population and external factors such as nutrients and temperature. It can also be used to describe the behavior of a capacitor in an electrical circuit.

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