8. Consider the differential equation $$ \frac{d y}{d t}=f(y) \text { where } f(2)=0, f^{\prime}(2)=0, f^{\prime \prime}(2)

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8. Consider the differential equation $$ \frac{d y}{d t}=f(y) \text { where } f(2)=0, f^{\prime}(2)=0, f^{\prime \prime}(2)<0 $$ Which type of equilibrium point is $$ y=2 $$ ? (a) sink (b) source (c) node

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To classify the equilibrium point $$ y = 2 $$ for the differential equation $$ \frac{dy}{dt} = f(y), $$ we analyze the behavior of $$ f(y) $$ near $$ y = 2 $$. ### Given: 1. $$ f(2) = 0 $$: This means $$ y = 2 $$ is an equilibrium point. 2. $$ f'(2) = 0 $$: The first derivative at the equilibrium is zero, indicating that linearization (using only the first derivative) is inconclusive for determining stability. 3. $$ f''(2) < 0 $$: The second derivative is negative, providing more insight into the behavior near $$ y = 2 $$. ### Analysis: Let's consider a small perturbation around the equilibrium point by setting $$ y = 2 + x $$, where $$ x $$ is small. Expanding $$ f(y) $$ around $$ y = 2 $$ using a Taylor series: $$ f(2 + x) \approx f(2) + f'(2)x + \frac{1}{2}f''(2)x^2 = \frac{1}{2}f''(2)x^2 $$ Since $$ f''(2) < 0 $$, we have: $$ \frac{dy}{dt} \approx -c x^2 $$ where $$ c > 0 $$. #### Behavior: - **For $$ y > 2 $$ ($$ x > 0 $$)**: $$ \frac{dy}{dt} < 0 $$, so $$ y $$ decreases toward 2. - **For $$ y < 2 $$ ($$ x < 0 $$)**: $$ \frac{dy}{dt} < 0 $$, so $$ y $$ continues to decrease away from 2. This indicates that $$ y = 2 $$ is **attracting** from one side ($$ y > 2 $$) and **repelling** from the other side ($$ y < 2 $$). Such an equilibrium is termed **semi-stable**. ### Conclusion: Among the given options: - **Sink** typically refers to an equilibrium that is attracting on all sides. - **Source** refers to an equilibrium that is repelling on all sides. - **Node** is generally used in higher-dimensional systems. However, since the equilibrium at $$ y = 2 $$ is attracting from one side and repelling from the other (semi-stable), the best available choice is: **Answer:** (a) sink $$ y = 2 $$ is a **sink** equilibrium point.

Consider the differential equation

Which type of equilibrium point is ?
(a) sink
(b) source
© node

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Consider the differential equation Which type of equilibrium point is ? (a) sink (b) source © node