How do I determine equilibrium solutions and stability for a non-linear ODE?

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    Non-linear Ode
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SUMMARY

The discussion focuses on determining equilibrium solutions and stability for the non-linear ordinary differential equation (ODE) given by y' = y - y². Participants clarify that equilibrium solutions occur when y' = 0, leading to fixed points at y = 0 and y = 1. Stability is assessed by analyzing the direction field and the behavior of the function f(y) = y - y², where arrows indicate stability: if arrows point towards a fixed point, it is stable; if they point away, it is unstable. Additionally, calculating the derivative f'(y) at fixed points provides further insight into stability.

PREREQUISITES
  • Understanding of differential equations, specifically non-linear ODEs.
  • Familiarity with direction fields and isoclines.
  • Knowledge of equilibrium solutions and their significance in dynamic systems.
  • Ability to compute derivatives and analyze their implications for stability.
NEXT STEPS
  • Learn how to construct direction fields for various non-linear ODEs.
  • Study the concept of isoclines and their role in analyzing differential equations.
  • Explore the stability analysis of equilibrium points using the first derivative test.
  • Investigate additional methods for solving non-linear ODEs, such as phase plane analysis.
USEFUL FOR

Students and educators in differential equations, mathematicians interested in stability analysis, and anyone seeking to deepen their understanding of non-linear dynamic systems.

fzksfun
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Hey Guys,

I am really confused about the first problem on my first problem set in Diff Eq (not auspicious is it? Oh well...)

Draw the Direction field y' = y -y^2. Identify Isoclines and any equlibrium solutions.

I don't understand how to approach this problem because doesn't the y^2 term make the equation non-linear? Also, my professor mentioned that equilibrium solutions could be stable or unstable without telling us what that entailed so can you guys please enlighten me with that , also?

Thank you so much in advance.
 
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You should be able to read your book to find that to figure out eq. points, you set your y' = 0 and solve. Then there's a variety of ways to figure out stability, you can plot your f(y) = y - y^2 and then obviously where it crosses the horizontal axis, those are your fixed points. When your graph is above the horiz. axis, draw an arrow point right "-->" when it's below, draw an arrow point left "<---". Now if both your arrows point towards your fixed point then it's stable, otherwise unstable. You can also find f'(y) and plug in your fixed points.
 

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