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 values of curves:
Initial conditions of curves:
Number of curves:
Speed profiles of curves:
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, set $n_c$, the number of curves to 3 and drag the left points of each curve to set the correct initial condition. Click the curve to cycle between three different speed profiles (either “speed up,” “speed up, then slow down,” or “slow down”). Then, drag the second point on each curve so that the solution ends at the correct value.
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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 values of curves:
Initial conditions of curves:
Number of curves:
Speed profiles of curves:
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 the solution online, set $n_c$, the number of curves to 3 and drag the left points of each curve to set the correct initial condition. Click the curve to cycle between three different speed profiles (either “speed up,” “speed up, then slow down,” or “slow down”). Then, drag the second point on each curve so that the solution ends at the correct value.
Hide help