Glossary

asymptotically semi-stable solution

[latex]y=k[/latex] if it is neither asymptotically stable nor asymptotically unstable

asymptotically stable solution

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

asymptotically unstable solution

[latex]y=k[/latex] if there exists [latex]\epsilon >0[/latex] such that for any value [latex]c\in \left(k-\epsilon ,k+\epsilon \right)[/latex] the solution to the initial-value problem [latex]{y}^{\prime }=f\left(x,y\right),y\left({x}_{0}\right)=c[/latex] never approaches [latex]k[/latex] as [latex]x[/latex] 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 [latex]y=c[/latex], where [latex]c[/latex] 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 [latex]h[/latex] that is added to the [latex]x[/latex] value at each step in Euler’s Method