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March 31st, 2015, 10:30 PM  #1 
Member Joined: Apr 2014 From: australia Posts: 68 Thanks: 32  Eulers Method
hello, need some guidance with this question: Consider the initial value problem; dy/dt = (cosy + 3) / y, y(0)=0.5 a) Use Euler's method with 5 steps to estimate y(0.1) up to three decimal points. Answer: I am confident with this part; y1= 0.655 y2 = 0.771 y3 = 0.867 y4 = 0.951 y5 = 1.027 b) Sketch the direction field in the domain 0 ≤ t ≤ 0.1, 0.2 ≤ y ≤ 1.4. ? c) Determine whether the approximate solution overestimates or underestimates the exact solution. ?? Any help would be appreciated. 
April 1st, 2015, 06:51 AM  #2 
Math Team Joined: Dec 2013 From: Colombia Posts: 6,965 Thanks: 2282 Math Focus: Mainly analysis and algebra 
For the direction field, you pick pairs $(x,y)$ and at each point mark the gradient of the function. You can then sketch some solutions by following where the gradients lead. For the last part I think you should look at the solutions you sketched above. 

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