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

In summary, the conversation is about solving a problem in Differential Equations involving the equation y' = y - y^2, identifying Isoclines and equilibrium solutions, and understanding stability. The speaker is confused about the non-linearity of the equation and how to approach it. The other person explains that to find equilibrium solutions, you set y' = 0 and solve, and to determine stability you can plot f(y) = y - y^2 and see where it crosses the horizontal axis. If both arrows point towards the fixed point, it is stable, otherwise it is unstable. F'(y) can also be used to determine stability.
  • #1
fzksfun
3
0
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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  • #2
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.
 

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

What is a non-linear ODE?

A non-linear ODE (ordinary differential equation) is a mathematical equation that involves functions and their derivatives, where the derivatives are not proportional to the functions themselves. In other words, the dependent variable and its derivatives are not directly related in a simple way.

What is the difference between a linear and a non-linear ODE?

In a linear ODE, the dependent variable and its derivatives have a proportional relationship, meaning that the coefficients in the equation are constants. In a non-linear ODE, the coefficients can be functions of the dependent variable, making the equation more complex and difficult to solve.

What are some real-world applications of non-linear ODEs?

Non-linear ODEs are used in various fields of science and engineering to model complex systems and phenomena. Some examples include population growth, chemical reactions, fluid dynamics, and electrical circuits.

How do you solve a non-linear ODE?

Solving a non-linear ODE can be challenging and often requires the use of advanced mathematical techniques such as numerical methods or approximations. In some cases, an exact analytical solution may not be possible, and numerical approximations are used instead.

What are the limitations of using non-linear ODEs?

Non-linear ODEs can be difficult to solve and may not always have an exact solution. In addition, they may not accurately represent complex systems due to the simplifications and assumptions made in the modeling process. It is important to carefully consider the assumptions and limitations of using non-linear ODEs in any application.

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