6. Consider the initial value problem dy/ dt = t ^2 y , y(1) = 1...

6. Consider the initial value problem dy/ dt = t ^2 y , y(1) = 1 .

(a) Use Euler’s method (by hand) to approximate the solution y(t) at t = 2 using ∆t = 1, ∆t = 1/2 and ∆t = 1/4 (use a calculator to approximate the answer for the smallest ∆t). Report your results in a table listing ∆t in one column, and the corresponding approximation of y(2) in the other.

(b) The direction field of dy/dt = t ^2y is shown at right. In this figure, add a sketch of ◦ the exact solution y(t) to the given initial value problem, for t ∈ [1, 2] ◦ the numerical approximation using Euler’s method that you found above (indicate the solution obtained at each step, and a line showing how it is obtained)

Expert Solution

Colby Messinger answered 6 months ago

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