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 ( t ) = 4 arctan t and told that routine computations will show that x ( 0 ) = 0 and x ( 1 ) = π.
I must determine a differential equation for x ( t ) of the form x ( t ) = f ( t , x ).
I must then use f to approximate π.
I will use Euler's Method to find minimum number of iterations ( n) needed to yield the approximation π 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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Answers (1)

Kasey Bird
Answered 2022-08-13 Author has 13 answers
I think you are expected to divide [0,1] into n intervals, use Euler's method to integrate 1 1 + t 2 over the interval and see what you get for x=1. For n=5, you would compute x ( 1 5 ) = x ( 0 ) + 1 5 x ( 0 ) = 0 + 1 5 1 = 0.2, x ( 2 5 ) = x ( 1 5 ) + 1 5 x ( 1 5 ) 0.2 + 1 5 0.961538 0.392308 ,, continue stepping until you get to x(1), and compare it with π 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 ( 1 ) 0.809981, an error of about 0.025. Since the global truncation error goes as 1 n , we would expect to need 10 times more terms to get the error to be below 0.0025 so the error in π 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.
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First Ten Values:
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