Solve the dynamical system graphically for both the case of exponential growth as follows.
First, pick a value of $a$ that will lead to exponential growth. For that value of $a$, plot the right hand side of the dynamical system as a function of $z$. (I.e., plot the function $f(z)=az$, where you plot $z$ on the horizontal axis and $f$ on the vertical axis.)
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sign of slope:
vertical intercept:
Using this graph to guide you, sketch the solution of $z$ versus $t$ on the below graph. Sketch solutions for initial conditions $c=1$, $c=0$, and $c=-1$.
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Final value:
Initial conditions:
Speed profile:
Even though for some initial conditions, $z(t)$ is decreasing, we still call it exponential growth, as it is shooting off to large, negative values.
Hint
See
Solving single autonomous differential equations using graphical methods.
To plot the solution online, for each initial condition, enable the corresponding curve and select the correct option for how the speed of the solution changes (either “speed up,” “speed up, then slow down,” or “slow down”). Then, drag the points on the curve so that the solution begins and ends at the correct values. (If the solution is a constant, selecting “slow down” will work.) Use curve A for initial condition $c=1$, curve B for $c=0$ and curve C for $c=-1$.
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Next, pick a value of $a$ that will lead to exponential decay and plot the right hand side of the dynamical system as a function of $z$.
Feedback from applet
sign of slope:
vertical intercept:
Using this graph to guide you, sketch the solution of $z$ versus $t$ on the below graph. Sketch solutions for initial conditions $c=4$, $c=0$, and $c=-4$.
Feedback from applet
Final value:
Initial conditions:
Speed profile:
Even though for some initial conditions, $z(t)$ is increasing, we still call it exponential decay, as the solution is decaying toward zero.
Hint
To plot solution online, for each initial condition, enable the corresponding curve and select the correct option for how the speed of the solution changes (either “speed up,” “speed up, then slow down,” or “slow down”). Then, drag the points on the curve so that the solution begins and ends at the correct values. (If the solution is a constant, selecting “slow down” will work.) Use curve A for initial condition $c=4$, curve B for $c=0$ and curve C for $c=-4$.
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