150_pdfsam_math 54 differential equation solutions odd

150_pdfsam_math 54 differential equation solutions odd - x...

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Chapter 3 of the improved Euler’s method subroutine are F = f ( x, y )=1 y + y 3 , G = f ( x + h, y + hF )=1 ( y + hF )+( y + hF ) 3 . From Step 4 of the improved Euler’s method subroutine with x =0 , y =0,and h =1 ,weget x = x + h =0+1=1 , y = y + h 2 ( F + G )=0+ 1 2 ± 1+(1 1+1 3 ) ² =1 . Thus, φ (1) y (1; 1) = 1 . The algorithm (Step 1 of the improved Euler’s method subroutine) next sets h =2 1 =0 . 5. The inputs to the subroutine are x =0 , y =0, c =1 ,and N = 2. For Step 3 of the subroutine we have F =1 0+0=1 , G =1 [0 + 0 . 5(1)] + [0 + 0 . 5(1)] 3 =0
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Unformatted text preview: x = 0 + 0 . 5 = 0 . 5 , y = 0 + 0 . 25(1 + 0 . 625) = 0 . 40625 . Thus the approximate value of the solution at 0 . 5 is 0 . 40625. Next we repeat Step 3 with x = 0 . 5 and y = 0 . 40625 to obtain F = 1 − . 40625 + (0 . 40625) 3 = 0 . 6607971 , G = 1 − [0 . 40625 + 0 . 5(0 . 6607971)] + [0 . 40625 + 0 . 5(0 . 6607971)] 3 ≈ . 6630946 . In Step 4 we compute x = 0 . 5 + 0 . 5 = 1 , 146...
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