296_pdfsam_math 54 differential equation solutions odd

296_pdfsam_math 54 differential equation solutions odd -...

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Unformatted text preview: Chapter 5 Here f1 (t, x1 , x2 , x3 ) = x2 , f2 (t, x1 , x2 , x3 ) = x3 , f3 (t, x1 , x2 , x3 ) = t − x3 − x2 . 1 Since we are computing the approximations for c = 1, the initial value for h in Step 1 of the algorithm in Appendix E of the text is h = (1 − 0)2−0 = 1. The equations in Step 3 are k1,1 = hf1 (t, x1 , x2 , x3 ) = hx2 , k2,1 = hf2 (t, x1 , x2 , x3 ) = hx3 , k3,1 = hf3 (t, x1 , x2 , x3 ) = h t − x3 − x2 , 1 k1,1 k2,1 h k1,2 = hf1 t + , x1 + , x2 + , x3 + 2 2 2 k1,1 k2,1 h , x2 + , x3 + k2,2 = hf2 t + , x1 + 2 2 2 k3,1 2 k3,1 2 k2,1 , 2 k3,1 = h x3 + , 2 = h x2 + =h t+ k3,1 k1,1 h − x3 − − x1 + 2 2 2 2 k1,1 k2,1 k3,1 h , x2 + , x3 + k3,2 = hf3 t + , x1 + 2 2 2 2 h k1,3 = hf1 t + , x1 + 2 h k2,3 = hf2 t + , x1 + 2 k3,3 = hf3 k1,2 , x2 + 2 k1,2 , x2 + 2 k2,2 , x3 + 2 k2,2 , x3 + 2 k3,2 2 k3,2 2 , k2,2 , 2 k3,2 = h x3 + , 2 = h x2 + k3,2 k1,2 h − x1 + = h t + − x3 − 2 2 2 2 k1,2 k2,2 k3,2 h , x2 + , x3 + t + , x1 + 2 2 2 2 , k1,4 = hf1 (t + h, x1 + k1,3 , x2 + k2,3 , x3 + k3,3 ) = h (x2 + k2,3 ) , k2,4 = hf2 (t + h, x1 + k1,3 , x2 + k2,3 , x3 + k3,3 ) = h (x3 + k3,3 ) , k3,4 = hf3 (t + h, x1 + k1,3 , x2 + k2,3 , x3 + k3,3 ) = h t + h − x3 − k3,3 − (x1 + k1,3 )2 . Using the starting values t0 = 0, a1 = 1, a2 = 0, and a3 = 1, we obtain the ﬁrst approximations x1 (1; 1) = 1.29167 , x2 (1; 1) = 0.28125 , 292 ...
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This note was uploaded on 03/29/2010 for the course MATH 54257 taught by Professor Hald,oh during the Spring '10 term at University of California, Berkeley.

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