Answer :
Answer:
y(2) = 8.248
Step-by-step explanation:
y(2) is the value of y when x = 2.
f(x,y) = y' = x sine y
[tex]y_{0} = 8, x_{0} = 0[/tex]
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[tex]y_{1} = y_{0} + 0.25*f(x_{0},y_{0}) = 8 + 0 = 8[/tex]
[tex]x_{1} = x_{0} + dx = 0.25[/tex]
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[tex]y_{2} = y_{1} + 0.25*f(x_{1},y_{1}) = 8+0.25*f(0.25,8) = 8 +8.7*10^{-3} = 8.008[/tex]
[tex]x_{2} = x_{1} + dx = 0.25 + 0.25 = 0.50[/tex]
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[tex]y_{3} = y_{2} + 0.25*f(x_{2},y_{2}) = 8.008+0.25*f(0.50,8.008) = 8.008 + 0.02 = 8.028[/tex]
[tex]x_{3} = x_{2} + dx = 0.50 + 0.25 = 0.75[/tex]
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[tex]y_{4} = y_{3} + 0.25*f(x_{3},y_{3}) = 8.028+0.25*f(0.75,8.028) = 8.028 + 0.026 = 8.054[/tex]
[tex]x_{4} = x_{3} + dx = 0.75 + 0.25 = 1[/tex]
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[tex]y_{5} = y_{4} + 0.25*f(x_{4},y_{4}) = 8.054+0.25*f(1,8.054) = 8.054 + 0.035 = 8.089[/tex]
[tex]x_{5} = x_{4} + dx = 1 + 0.25 = 1.25[/tex]
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[tex]y_{6} = y_{5} + 0.25*f(x_{5},y_{5}) = 8.089+0.25*f(1.25,8.089) = 8.089 + 0.044 = 8.133[/tex]
[tex]x_{6} = x_{5} + dx = 1.25 + 0.25 = 1.50[/tex]
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[tex]y_{7} = y_{6} + 0.25*f(x_{6},y_{6}) = 8.133+0.25*f(1.50,8.133) = 8.133 + 0.053 = 8.186[/tex]
[tex]x_{7} = x_{6} + dx = 1.50 + 0.25 = 1.75[/tex]
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[tex]y_{8} = y_{7} + 0.25*f(x_{7},y_{7}) = 8.186+0.25*f(1.75,8.186) = 8.186+ 0.062 = 8.248[/tex]
[tex]x_{8} = x_{7} + dx = 1.75 + 0.25 = 2[/tex]
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So, after 8 iterations at the Euler's method, we arrive at the moment x = 2. At this moment, we have that y = 8.248. So, y(2) = 8.248