-s1.2.4 Find the equilibrium solution y_e

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• karush
In summary, an equilibrium solution is a constant value of a variable in a system that is balanced by forces or factors, resulting in no net change. It is found by setting the derivative of the variable equal to zero and solving for the value. This is important in understanding and predicting the behavior of a system and identifying stable and unstable points. Multiple equilibrium solutions can exist in a system when different forces balance each other out at different points. The equilibrium solution has a significant impact on real-world systems, aiding in the study and decision-making in various fields.
karush
Gold Member
MHB
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$\textsf{Consider the differential equation$\displaystyle \frac{dy}{dt}=ay-b$}$

(a) Find the equilibrium solution $y_e$
rewrite as
$y'-ay=b$
$\displaystyle -\exp\int a \, da=e^{a^{2}/2}$
$\color{red}{y_e=b/a}$

(b) Let $Y(t)=y-y_e$; thus $Y(t)$ is the deviation from the equilibrium solution.
Find the differential equation satisfied by $Y(t)$.
?
$\color{red}{Y' = aY}$
ok stopped in my tracks.. red is book answer

Last edited:
karush said:
$\textsf{Consider the differential equation$\displaystyle \frac{dy}{dt}=ay-b$}$

(a) Find the equilibrium solution $y_e$
rewrite as
$y'-ay=b$
$\displaystyle -\exp\int a \, da=e^{a^{2}/2}$
$\color{red}{y_e=b/a}$

(b) Let $Y(t)=y-y_e$; thus $Y(t)$ is the deviation from the equilibrium solution.
Find the differential equation satisfied by $Y(t)$.
?
$\color{red}{Y' = aY}$
ok stopped in my tracks.. red is book answer

Okay, we are given:

$$\displaystyle \d{y}{t}=ay-b$$

Any equilibrium solutions are found from:

$$\displaystyle \d{y}{t}=0$$

$$\displaystyle ay-b=0\implies y_e=\frac{b}{a}\quad\checkmark$$

Next, we are given:

$$\displaystyle Y(t)=y-y_e$$

This implies:

$$\displaystyle \d{Y}{t}=\d{y}{t}\implies \d{Y}{t}=ay-b=a(Y+y_e)-b=aY+a\frac{b}{a}-b=aY$$

Make sense?

 $\displaystyle ay-b=0\implies y_e=\frac{b}{a}$ uhmm how did $y$ become $y_e$

Last edited:
karush said:
 $\displaystyle ay-b=0\implies y_e=\frac{b}{a}$ uhmm how did $y$ become $y_e$

The solution we are finding in this case is $$y_e$$, since we have set the derivative to zero.

1. What is an equilibrium solution?

An equilibrium solution is the value of a variable in a system that remains constant over time. In other words, it is the point at which the forces or factors affecting the variable are balanced, resulting in no net change.

2. How is the equilibrium solution found?

The equilibrium solution is found by setting the derivative of the variable equal to zero and solving for the variable. This results in the value of the variable at which there is no change over time.

3. Why is finding the equilibrium solution important?

Finding the equilibrium solution is important because it helps us understand the behavior of a system and predict its future states. It also allows us to identify stable and unstable points in the system.

4. Can there be more than one equilibrium solution?

Yes, there can be multiple equilibrium solutions in a system. This can occur when there are different forces or factors that can balance each other out at different points, resulting in multiple stable points.

5. How does the equilibrium solution impact real-world systems?

The equilibrium solution has a significant impact on real-world systems as it helps us understand the stability of a system and predict its behavior. It is useful in fields such as economics, engineering, and ecology to study the behavior of complex systems and make informed decisions.

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