Exercise · ZW-PHASE-PLANE-001 · ★☆☆ Basic
For \(x'=x+y,\ y'=x-y\), what is the slope \(dy/dx\) of the trajectory through \((2,1)\)?
Choose one option.
Show answer
- C. \(1/3\)
Explanation
At \((2,1)\): \(x'=2+1=3\) and \(y'=2-1=1\), so \(dy/dx=y'/x'=1/3\). Inverting the quotient gives the typical error \(3\).
Review the concept: Phase plane
Coordinate plane \((x,y)\) of an autonomous system whose velocities \(f\) and \(g\) do not depend on \(t\); each solution is drawn there as a curve, its trajectory.
\[x'=f(x,y),\quad y'=g(x,y)\;\Longrightarrow\;\dfrac{dy}{dx}=\dfrac{g(x,y)}{f(x,y)}\quad(f\neq0)\]
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