Solving the Non-Linear ODE: x+ x^2*y+x^3*y^2

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Discussion Overview

The discussion centers around solving the non-linear ordinary differential equation (ODE) given by dy/dx = x + x^2*y + x^3*y^2. Participants explore various methods and approaches to tackle this equation, which is identified as a Riccati equation. The scope includes theoretical and practical considerations in solving ODEs, particularly in the context of financial mathematics.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Homework-related

Main Points Raised

  • Some participants identify the equation as a Riccati equation and suggest a substitution of v = x^3*y to simplify it.
  • Others propose that the equation may require numerical methods for a practical solution, given its complexity.
  • One participant notes that analytical solutions could involve confluent hypergeometric functions, but expresses concern about the relevance of this approach to the original poster.
  • There is a suggestion that the original poster should provide details of their attempts to solve the equation to facilitate more targeted assistance.

Areas of Agreement / Disagreement

Participants generally agree that the equation is a Riccati equation and can be approached through various methods. However, there is no consensus on the best method to solve it, with differing opinions on the feasibility of analytical versus numerical solutions.

Contextual Notes

Some limitations include the need for specific assumptions in the proposed substitutions and the complexity of the analytical solution involving special functions, which may not be suitable for all participants.

scizj
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Hi, All:


I am taking a financial maths course and I encounter the following ODE:

dy/dx = x+ x^2*y+x^3*y^2


I have tried many methods but cannot solve it.

Can anyone help me? Thanks.
 
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scizj said:
Hi, All:


I am taking a financial maths course and I encounter the following ODE:

dy/dx = x+ x^2*y+x^3*y^2


I have tried many methods but cannot solve it.

Can anyone help me? Thanks.


The equation is non linear ODE, you probably have to use a computer to solve it for you

have read about linearity of ODEs
 
scizj said:
Hi, All:


I am taking a financial maths course and I encounter the following ODE:

dy/dx = x+ x^2*y+x^3*y^2


I have tried many methods but cannot solve it.

Can anyone help me? Thanks.

You will need to show us how and what you did in trying to solve it first before one can ascertain if you're right or wrong.
Hint: Think of letting v= x^2 y.
 
I am taking a financial maths course and I encounter the following ODE:
dy/dx = x+ x^2*y+x^3*y^2
I have tried many methods but cannot solve it.
Can anyone help me? Thanks.
This non-linear ODE can be solved, thanks to the general method for solving Riccati equations.
Nevertheless, the analytical solving is rather ardous in the present case : It would involve confluent hypergeometric functions. We could do it, but I am afraid that there would be of no interest for you. Probably, the use of numerical methods of computation would be more convenient in practice.
 

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