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Unformatted text preview: Approximation Consider again the FOOIVP y ( t ) = f ( t,y ( t )) , y ( t ) = y . This covers systems and higher order ODE. If one of our four arrows does not shoot it down, well try numerical approximation of the solution (Note the power series so lutions are practically approximations). Well talk about three approximation schemes 1) the Euler Method ; 2) the Improved Euler Method , also called the Heun Method ; 3) the RungeKutta Method . All three methods consist in a prescrip tion to move from the starting point ( t ,y ) to the next point ( t 1 ,y 1 ) , from there to the next point ( t 2 ,y 2 ) , then to ( t 3 ,y 3 ) , ... . We connect these points by straight lines, ob taining a polygon, which we hope is an approximation to the real solution. The Euler Method The rule to get from one point ( t,y ) to the next point ( t, y ) is this: Choose a Step Size h . Then set t def = t + h and y def = y + f ( t,y ) h t y = t + h y + f ( t,y ) h 2 ( t,y ) t y ( t n ,y n ) ( t 1 ,y 1 ) t n ( t...
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 Spring '08
 Turner

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