366 lab8

366 lab8 - 2 (y + 1) 7.-2 2 4 6 8 10 2 4 6 8 10 t y y ' =...

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0 0.5 1 1.5 2 2.5 3 0 1 2 3 4 5 6 7 8 t y y ' = y + t 1. Slope = f(2,1) = 2+1=3 When t= 2 y=1, when t = 3, y = 4 Slope = (4-1)/(3-1) = 3 2. k1 = f(1,0.5) =1+0.5 = 1.5 k2 = f(2,2) =2+2 = 4 m0 = (1.5+4)/2 = 2.75 m* = (3.25-0.5)/(2-1) = 2.75 = k t1 = 1+1 = 2 , y1 = 2.75*1+0.5 = 3.25
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0 0.5 1 1.5 2 2.5 3 0 1 2 3 4 5 6 7 8 t y y ' = y + t It agrees with the graph. 3. m0 = f(0,0) = 0 m1 = f(0.5,0) = 0.5 y2 = 0.5*0.5 = 0.25 m2 = f(1,0.25) = 1.25
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0 0.5 1 1.5 2 2.5 3 0 1 2 3 4 5 6 7 8 t y y ' = y + t 4. t0 = 0, y0 = 0.5, k0 =0.5 t1 = 0.5 , y1 = 0.5*0.5=0.25,k1 = 0.5+0.25 = 0.75
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-2 0 2 4 6 8 10 -4 -3 -2 -1 0 1 2 3 4 t y y ' = (3 - y) (y + 1) t2 = 1, y2 = 0.75*0.5+0.25 = 0.625 , k2 = 1.625 t3 = 1.5, y3 = 1.625*0.5+0.625 = 1.4375 m0 = (0.5+0.75)/2 = 0.625 m1 = (0.75+1.625)/2 = 1.1875 m2= (1.625+1.4375)/2 = 1.53125 5. Euler : 4.3 Improved Euler : 0.3 6.
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-2 0 2 4 6 8 10 0 2 4 6 8 10 t y y ' = (3 - y)
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Unformatted text preview: 2 (y + 1) 7.-2 2 4 6 8 10 2 4 6 8 10 t y y ' = (3 - y) 2 (y + 1) Step size = 0.1 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 2 4 6 8 10 t y y ' = (3 - y) 2 (y + 1) 9. Maximum errors: Euler : >8 RK2 : >8 RK4: >8 10. With overly large step sizes, the inaccuracies generated by the three methods are all very large. With smaller step sizes, Euler is the most inaccurate method. Because RK uses average values, the answer can be more accurate. I cannot be certain that the graph produced by dfield is correct. Although dfield uses a small step size, there is still inaccuracy. Maybe these small inaccuracy can be ignored, it still exists. 11....
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This note was uploaded on 05/15/2011 for the course MA 366 taught by Professor Cho during the Spring '08 term at Purdue.

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366 lab8 - 2 (y + 1) 7.-2 2 4 6 8 10 2 4 6 8 10 t y y ' =...

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