Euler's formula & Euler's identity
11:27
Khan AcademySal Khan
Euler's formula & Euler's identity
Euler's Formula, e^(ix) = cos(x) + i*sin(x), and Euler's Identity, e^(iπ) + 1 = 0, are incredible math discoveries. They connect e, i, and pi through Maclaurin series of cos(x), sin(x), and e^x. These formulas reveal a deep, mysterious connection between important numbers in math.
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(a) Program a calculator or computer to use Euler's method to compute y(1), where y(x) is the solution of the initial-value problem (d y)/(d x) 3 x^2 y=6 x^2 y(0)=3 for each of the given step sizes. (i) h=1 (ii) h=0.1 (iii) h=0.01 (iv) h=0.001 (b) Verify that y=2 e^-x^3 is the exact solution of the differential equation. (c) Find the errors in using Euler's method to compute y(1) with the step sizes in part (a). What happens to the error when the step size is divided by 10 ? | Numerade
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(a) Program a calculator or computer to use Euler's method to compute y(1), where y(x) is the solution of the initial-value problem (d y)/(d x) 3 x^2 y=6 x^2 y(0)=3 for each of the given step sizes. (i) h=1 (ii) h=0.1 (iii) h=0.01 (iv) h=0.001 (b) Verify that y=2 e^-x^3 is the exact solution of the differential equation. (c) Find the errors in using Euler's method to compute y(1) with the step sizes in part (a). What happens to the error when the step size is divided by 10 ? | Numerade
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