Sensitivity to initial conditions is well illustrated by a little target practice with your numerical solver. Enter $x' = x^2 - t$ into your numerical solver, and then experiment with initial conditions at $t_0 = 0$ until the solution with $x(t_0,\ x_0)$ hits the target $(5,\ 0)$.

I don't fully understand what the goal is here or what is being asked. I'm uncertain of what the textbook means by "numerical solver." In other contexts, I understanding it to mean a numerical method algorithm, but it seems to be referring to software, perhaps software that graphs the slope field of the given ODE? Which of the following qualify as numerical solvers, and for each that does, outline the steps one would go through to complete the exercise and understand the concept being illustrated?

$\bullet$ TI-84
$\bullet$ Desmos
$\bullet$ MATLAB

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    $\begingroup$ None of these is a "numerical solver", but I suspect they all contain numerical solvers. A numerical solver takes a differential equation $dx/dt = \ldots$, initial condition $x(t_0) = x_0$, and number $t_1$ and (approximately) computes $x(t_1)$ for a solution of the initial value problem. $\endgroup$ – Robert Israel Apr 12 at 19:22

For example, you might start with the initial condition $x(0) = 0$. Your numerical solver might tell you $x(5) = -2.18278513266442$. OK, try something bigger, say $x(0)=1$. The numerical solver tells you it encountered a singularity at around $t = 1.1$ (i.e. it seems the solution goes off to $\infty$ before you get to $5$). Next try $x(0)=0.5$. You get $x(5) = -2.18277934097677$. Next maybe $x(0) = 0.75$.... Somewhere between the values where the solution at $t=5$ is around $-2.18$ and the values where it goes to $\infty$ you should find an $x_0$ that gives you something close to $x(5)=0$.


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