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March 31st, 2015, 10:30 PM   #1
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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.
Thanks from Yury Stepanyants
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April 1st, 2015, 06:51 AM   #2
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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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