Math 310 Q6 Sol - MATH 310, Fall 2011: Differential...

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MATH 310, Fall 2011: Differential Equations Simon Fraser University Quiz 6: Solutions 1. Consider the initial value problem y 0 + 2 y 2 = 3 t + e y , y (1) = 0 . Apply one step of Euler’s method with step size h = 0 . 01 to find an approxi- mation y 1 to y (1 . 01). Write an arithmetic expression for y 2 , the second Euler iterate (that is, substi- tute the appropriate numerical values; you do not need to evaluate y 2 ). Solution: Euler’s method is appropriate for DEs of the form y 0 = f ( t,y ) ; in this case (solving for y 0 ) we have f ( t,y ) = - 2 y 2 + 3 t + e y . The method has the form y n +1 = y n + hf ( t n ,y n ) for n = 0 , 1 , 2 ,... . The initial condition y (1) = 0 implies t 0 = 1 , y 0 = 0 ; the step size is h = 0 . 01 . Thus we compute y 1 = y 0 + hf ( t 0 ,y 0 ) = y 0 + h ( - 2 y 2 0 + 3 t 0 + e y 0 ) = 0 + 0 . 01 ( - 2(0) 2 + 3(1) + e 0 ) = 0 . 01(0 + 3 + 1) = 0 . 04; this is an approximation to y ( t 1 ) , where t 1 = t 0 + h = 1 . 01 .
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Math 310 Q6 Sol - MATH 310, Fall 2011: Differential...

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