Glossary

asymptotically semi-stable solution

$y=k$ if it is neither asymptotically stable nor asymptotically unstable

asymptotically stable solution

$y=k$ if there exists $\epsilon >0$ such that for any value $c\in \left(k-\epsilon ,k+\epsilon \right)$ the solution to the initial-value problem ${y}^{\prime }=f\left(x,y\right),y\left({x}_{0}\right)=c$ approaches $k$ as $x$ approaches infinity

asymptotically unstable solution

$y=k$ if there exists $\epsilon >0$ such that for any value $c\in \left(k-\epsilon ,k+\epsilon \right)$ the solution to the initial-value problem ${y}^{\prime }=f\left(x,y\right),y\left({x}_{0}\right)=c$ never approaches $k$ as $x$ approaches infinity

direction field (slope field)

a mathematical object used to graphically represent solutions to a first-order differential equation; at each point in a direction field, a line segment appears whose slope is equal to the slope of a solution to the differential equation passing through that point

equilibrium solution

any solution to the differential equation of the form $y=c$, where $c$ is a constant

Euler’s Method

a numerical technique used to approximate solutions to an initial-value problem

solution curve

a curve graphed in a direction field that corresponds to the solution to the initial-value problem passing through a given point in the direction field

step size

the increment $h$ that is added to the $x$ value at each step in Euler’s Method