Need help solving this differential equation

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Discussion Overview

The discussion revolves around solving a specific differential equation, focusing on the methods of separation of variables and potential substitutions for integration. Participants explore various approaches to simplify and solve the integral derived from the equation.

Discussion Character

  • Technical explanation
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant presents the separated form of the differential equation but struggles to find a solution using non-trig substitution methods.
  • Another participant questions the initial formulation, noting the absence of an equals sign and requests the original equation for clarity.
  • A participant provides the original differential equation and describes their attempts at substitution, suggesting that a trigonometric substitution may be necessary but expressing uncertainty due to limited experience with such methods.
  • Another participant suggests that instead of manipulating the ODE, the focus should be on the integral, proposing a simplification and a potential path using trigonometric substitution and partial fractions.
  • Some participants express difficulty in progressing further after the suggested simplifications, indicating a shared sense of challenge regarding the complexity of the integral.
  • A later reply mentions that the integral is very complicated, involving a lengthy formula related to the roots of a 6th degree polynomial, and suggests using software like Maple or Mathematica for assistance.
  • One participant expresses interest in obtaining a solution now that the thread is outside the schoolwork forum, implying a desire for more detailed guidance.

Areas of Agreement / Disagreement

Participants generally agree on the complexity of the integral and the challenges faced in finding a solution. However, there is no consensus on the best approach to take, with differing opinions on whether to continue manipulating the ODE or focus on the integral.

Contextual Notes

Participants note the integral's complexity and the involvement of a 6th degree polynomial, which may limit the applicability of certain methods. There is also uncertainty regarding the effectiveness of trigonometric substitutions due to varying levels of experience among participants.

MegaFlyman
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I've separated the variables of this differential equation and end up with
dx/((a-x)^(1/2)*(b-c(x-d)^3/2)). I've tried finding the integral of this with non-trig substitution methods but cannot solve it. Any help would be appreciated.
 
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MegaFlyman said:
I've separated the variables of this differential equation and end up with
dx/((a-x)^(1/2)*(b-c(x-d)^3/2)).
I don't see an equation here. (Where's the equals sign?) Start by writing the original problem and show us what work you have.
 
The original diff eq is

dx/dy = b(a-x)^1/2 - c(a-x)^1/2 * (x-d)^3/2

Separating variables results in my original posted equation

dy = dx/((a-x)^1/2 * (b-c(x-d)^3/2))

I have tried the substitution, u = (a-x)^1/2, x = a-u^2, dx = -2udu. Which results in

dy = -2dx/(b-c((-u^2+a)-d)^3/2)

Any further non-trig substitutions does not help to simplify. I believe a trig substitution is required but I have little experience with trig subs. I have already put any many hours looking for a solution and would like to know if a trig sub could be used to solve this. Thanks
 
I don't think there's any point in manipulating the ODE. You have reduced it to an integral, so work with that. Your substitutions so far look good, but I believe you can simplify it to ##\frac{du}{A-(1-u^2)^{\frac32}}##. Substituting u = sin(θ) and expanding with partial fractions can get you to a sum of terms like ##\frac{d\theta}{W-cos(\theta)}##, but I don't know where to go from there.
 
I'm not seeing anywhere to go after that either. But thanks for the input.
 
MegaFlyman said:
I'm not seeing anywhere to go after that either. But thanks for the input.

The integral is VERY complicated. It involves a lengthy formula that uses the roots of a 6th degree polynomial whose coefficients are functions of a, b, c and d. I did the integral in Maple, but if you do not have access to Maple you could try to submit it to Mathematica. Wolfram Alpha failed to find the integral.
 
Glad to hear that there is a solution. Since this thread is no longer in the schoolwork forum, would you be more forthcoming with the solution.
 

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