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EXAMPLE OF ONE-STEP METHOD Consider solving
 

Summary: EXAMPLE OF ONE-STEP METHOD
Consider solving
y0 = y cos x; y(0) = 1
Imagine writing a Taylor series for the solution Y (x),
say initially about x = 0. Then
Y (h) = Y (0) + hY 0(0) +
h2
2
Y 00(0) +
h3
6
Y 000(0) +
We can calculate Y 0(0) = Y (0) cos(0) = 1. How do
we calculate Y 00(0) and higher order derivatives?
Y 0(x) = Y (x) cos(x)
Y 00(x) = Y (x) sin(x) + Y 0(x) cos(x)
Y 000(x) = Y (x) cos(x) 2Y 0(x) sin(x)+Y 00(x) cos x
Then Y (0) = 1, Y 0(0) = 1;and
Y 00(0) = Y (0) sin(0) + Y 0(0) cos(0) = 1
Y 000(0) = Y (0) cos(0) 2Y 0(0) sin(0)+Y 00(0) cos 0 = 0

  

Source: Atkinson, Kendall - Departments of Computer Science & Mathematics, University of Iowa

 

Collections: Computer Technologies and Information Sciences