The Jacobian matrix of the system $x ^ { \prime } = x ^ { 3 } - y ^ { 2 } + 4 , y ^ { \prime } = 5 x$ at the critical point $( 0,2 )$
A) $A = \left[ \begin{array} { c c }
0 & - 4 \
5 & 0
\end{array} \right]$
B) $A = \left[ \begin{array} { l l }
0 & 4 \
5 & 0
\end{array} \right]$
C) $A = \left[ \begin{array} { r r }
3 x ^ { 2 } & 2 y \
5 & 0
\end{array} \right]$
D) $A = \left[ \begin{array} { r r }
3 x ^ { 2 } & - 2 y \
5 & 0
\end{array} \right]$
E) $A = \left[ \begin{array} { r r }
3 x ^ { 2 } & - 2 y \
0 & 5
\end{array} \right]$

Question

The Jacobian matrix of the system $x ^ { \prime } = x ^ { 3 } - y ^ { 2 } + 4 , y ^ { \prime } = 5 x$ at the critical point $( 0,2 )$
A) $A = \left[ \begin{array} { c c } 
0 & - 4 \
5 & 0
\end{array} \right]$
B) $A = \left[ \begin{array} { l l } 
0 & 4 \
5 & 0
\end{array} \right]$
C) $A = \left[ \begin{array} { r r } 
3 x ^ { 2 } & 2 y \
5 & 0
\end{array} \right]$
D) $A = \left[ \begin{array} { r r } 
3 x ^ { 2 } & - 2 y \
5 & 0
\end{array} \right]$
E) $A = \left[ \begin{array} { r r } 
3 x ^ { 2 } & - 2 y \
0 & 5
\end{array} \right]$

Quizplus · Accepted Answer

The Jacobian matrix of a system $x' = f(x, y), y' = g(x, y)$ is given by $\left[ \begin{array}{cc} \frac{\partial f}{\partial x} & \frac{\partial f}{\partial y} \ \frac{\partial g}{\partial x} & \frac{\partial g}{\partial y} \end{array} \right]$. For the given system, $f(x, y) = x^3 - y^2 + 4$ and $g(x, y) = 5x$, so the Jacobian matrix at any point $(x, y)$ is $\left[ \begin{array}{cc} 3x^2 & -2y \ 5 & 0 \end{array} \right]$. Evaluating this at the critical point $(0, 2)$ gives $\left[ \begin{array}{cc} 0 & -4 \ 5 & 0 \end{array} \right]$.

The Jacobian Matrix of the System x′=x3−y2+4,y′=5xx ^ { \prime } = x ^ { 3 } - y ^ { 2 } + 4 , y ^ { \prime } = 5 xx′=x3−y2+4,y′=5x

The Jacobian Matrix of the System $x ^ { \prime } = x ^ { 3 } - y ^ { 2 } + 4 , y ^ { \prime } = 5 x$