Discussion Overview
The discussion revolves around solving the non-exact ordinary differential equation (ODE) given by $$(y-2x^2y)dx +xdy = 0$$. Participants explore methods for determining an integrating factor and discuss the separability of the equation, as well as the implications of using different techniques for solving it.
Discussion Character
- Technical explanation
- Debate/contested
- Mathematical reasoning
Main Points Raised
- One participant asserts that the ODE is in exact form but fails the exactness test, leading them to use the technique $$\frac{M_y-N_x}{N}$$ to find an integrating factor, resulting in $u(x)=-2x$.
- Another participant suggests that the ODE is separable and proposes factoring out the $y$ from the $dx$ term.
- A later reply emphasizes the need to exponentiate the integral of $(M_y-N_x)/N$, proposing $u(x)=e^{-x^2}$ as a different integrating factor.
- One participant agrees with the separability approach and provides a transformed version of the ODE after applying the integrating factor $e^{-x^2}$, claiming it becomes exact.
Areas of Agreement / Disagreement
Participants express differing views on the best approach to solve the ODE, with some advocating for the use of an integrating factor while others suggest treating it as separable. The discussion remains unresolved regarding the effectiveness of the initial method used by the first participant.
Contextual Notes
There are limitations in the assumptions made about the exactness of the ODE and the validity of the integrating factors proposed. The discussion does not resolve the mathematical steps involved in determining the correct integrating factor.