Question

In each of the Problems below, determine the critical point x = x⁰, and then classify its type and examine its stability by making the transformation x = x⁰ + u.
1. dx/ dt = [ 1 1 [ 2
x −
1 −1 ] 0 ]

2. dx/ dt = [ -2 1 [ -2
x −
1 −2 ] 1 ]

3. dx/ dt = [ -1 -1 [ -1
x −
2 −1 ] 5 ]

4. dx/ dt = [ 0 −β [ α
x − α, β, γ, δ > 0
δ 0 ] −γ ]

Answer 1

Answer:

\bf xy=1\iff y = \cfrac{1}{x}\iff y = x^{-1}\\\\
-----------------------------\\\\
\cfrac{dy}{dt}=-x^{-2}\cdot \cfrac{dx}{dt}\impliedby \textit{using the chain-rule}
\\\\\\
\left. \cfrac{dy}{dt}=-\cfrac{\frac{dx}{dt}}{x^2} \right|{
\begin{array}{llll}
x=2\\
\frac{dx}{dt}=-2
\end{array}}\implies \cfrac{dy}{dt}=-\cfrac{-2}{2^2}

Step-by-step explanation: