Difficult first order linear differential equation

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Discussion Overview

The discussion revolves around solving the first order linear differential equation given by $y'=\dfrac{x^3-y^3}{x-y}$. Participants explore various methods and solutions, including the use of computational tools and analytical approaches.

Discussion Character

  • Exploratory
  • Technical explanation
  • Mathematical reasoning

Main Points Raised

  • One participant seeks assistance in solving the differential equation and asks for methods to approach it.
  • Another participant mentions that a computational tool (Wolfram Development Platform) provides a complex solution involving Hermite functions and hypergeometric functions, prompting a request for context and previous attempts at solving the equation.
  • A different participant shares a specific solution derived from a differential equation solver, presenting a lengthy expression for $y(x)$ that includes constants and polynomial terms.
  • Another participant suggests a reduction of the original equation to a different form: $y'=x^2 + x y + y^2$.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the solution to the differential equation, with multiple approaches and interpretations presented. The discussion remains unresolved regarding the best method to solve the equation.

Contextual Notes

The complexity of the solutions and the reliance on computational tools highlight potential limitations in analytical approaches. The discussion also reflects varying levels of familiarity with differential equations among participants.

WMDhamnekar
MHB
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Hello, I want to solve the following differential equation. $y'=\dfrac{x^3-y^3}{x-y}$. How to solve it?
 
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Wolfram Development Platform (essentially Mathematica) gives a horrendous answer: extremely complicated with Hermite $H_n(x)$ functions and the hypergeometric $_1F_1$ function. In what context did this problem come up, and what have you tried?
 
Ackbach said:
Wolfram Development Platform (essentially Mathematica) gives a horrendous answer: extremely complicated with Hermite $H_n(x)$ functions and the hypergeometric $_1F_1$ function. In what context did this problem come up, and what have you tried?

I got the following answer from my differential equation solver.

\[y(x)=\frac{x^3}{6}(C_1(C_1+1)+3C_1+2)+\frac{x^5}{120}(C^3_1+C_1(74C_1+9)+31C_1+8)+C_1+C_1x+\frac{3C_1x^2}{2}+\frac{C_1x^4}{12}(C_1+4)+\mathcal{O}\]
 
Last edited by a moderator:
You can reduce the expression to this

y'=x^2 + x y + y^2
 

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