 # I am given the function x(t)=4arctant and told that routine computations will show that x(0)=0 and x(1)=pi.. I must determine a differential equation for x(t) of the form x′(t)=f(t,x). I must then use f to approximate pi. Marco Hudson 2022-08-12 Answered
I am given the function $x\left(t\right)=4\mathrm{arctan}t$ and told that routine computations will show that $x\left(0\right)=0$ and $x\left(1\right)=\pi$.
I must determine a differential equation for $x\left(t\right)$ of the form ${x}^{\prime }\left(t\right)=f\left(t,x\right)$.
I must then use $f$ to approximate $\pi$.
I will use Euler's Method to find minimum number of iterations ($n$) needed to yield the approximation $\pi \approx 3.1400$, such that $n-1$ iterations is strictly less than 3.1400.
How do I find $n$? It seems to me that it can't be found given the instruction set that I am provided.
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I think you are expected to divide [0,1] into n intervals, use Euler's method to integrate $\frac{1}{1+{t}^{2}}$ over the interval and see what you get for x=1. For n=5, you would compute $x\left(\frac{1}{5}\right)=x\left(0\right)+\frac{1}{5}{x}^{\prime }\left(0\right)=0+\frac{1}{5}\cdot 1=0.2$, $x\left(\frac{2}{5}\right)=x\left(\frac{1}{5}\right)+\frac{1}{5}{x}^{\prime }\left(\frac{1}{5}\right)\approx 0.2+\frac{1}{5}\cdot 0.961538\approx 0.392308,$, continue stepping until you get to x(1), and compare it with $\frac{\pi }{4}$. The problem I have is that the derivative is decreasing over the interval, so the forward Euler's method overestimates the function, while the problem assumes it underestimates the function. At n=10 I get $x\left(1\right)\approx 0.809981$, an error of about 0.025. Since the global truncation error goes as $\frac{1}{n}$, we would expect to need 10 times more terms to get the error to be below 0.0025 so the error in $\pi$ is less than 0.01 and that is what I find in Excel. You would need to look more closely to find the exact n requiblack.