13.Using three terms in the Taylor series given in equation (12) and taking h = 0.1, determine approximate values of the solution of the illustrative example y′ = 1 − t + 4y, y(0) = 1 at t = 0.1 and 0.2. Compare the results with those using the Euler method and with the exact values. Hint: If y′ = f(t, y), what is y″? In each of Problems 14 and 15, a.Estimate the local truncation error for the Euler method in terms of the solution y = ϕ (t). b.Obtain a bound for en + 1 in terms of t and ϕ (t) that is valid on the interval 0 ≤ t ≤ 1. c.By using a formula for the solution, obtain a more accurate error bound for en + 1. d.For h = 0.1 compute a bound for e1 and compare it with the actual error at t = 0.1. e.Compute a bound for the error e4 in the fourth step. 14.y′=2y−1,y(0)=1
13.Using three terms in the Taylor series given in equation (12) and taking h = 0.1, determine approximate values of the solution of the illustrative example y′ = 1 − t + 4y, y(0) = 1 at t = 0.1 and 0.2. Compare the results with those using the Euler method and with the exact values.
Hint: If y′ = f(t, y), what is y″?
In each of Problems 14 and 15,
a.Estimate the local truncation error for the Euler method in terms of the solution y = ϕ (t).
b.Obtain a bound for en + 1 in terms of t and ϕ (t) that is valid on the interval 0 ≤ t ≤ 1.
c.By using a formula for the solution, obtain a more accurate error bound for en + 1.
d.For h = 0.1 compute a bound for e1 and compare it with the actual error at t = 0.1.
e.Compute a bound for the error e4 in the fourth step.
14.y′=2y−1,y(0)=1
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