(a) Derive the Gaussian quadrature formula of the form fh f(x)dx = a f(h) + bf (2h) + cf (3h). Find the order of convergence of this formula. (b) Apply the formula you found on part (a) to Xi+k = x₁ + fti+k f(t, x(t))dt, to find an approximate formula for the numerical solution of the IVP x(t) = f(t, x(t)), x(to) = xo. (c) Use the IVP x (t) = 10 x(t) + 11t-5t²-1, x(0) = 0 to test the validity of the formula on the interval [0, 2] with h = 0.25. [Hint: the exact solution is x(t) = (½) t²-t.
(a) Derive the Gaussian quadrature formula of the form fh f(x)dx = a f(h) + bf (2h) + cf (3h). Find the order of convergence of this formula. (b) Apply the formula you found on part (a) to Xi+k = x₁ + fti+k f(t, x(t))dt, to find an approximate formula for the numerical solution of the IVP x(t) = f(t, x(t)), x(to) = xo. (c) Use the IVP x (t) = 10 x(t) + 11t-5t²-1, x(0) = 0 to test the validity of the formula on the interval [0, 2] with h = 0.25. [Hint: the exact solution is x(t) = (½) t²-t.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![-4h
(a) Derive the Gaussian quadrature formula of the form fh f(x) dx = a f(h) + bf (2h) + cf (3h). Find
the order of convergence of this formula.
(b) Apply the formula you found on part (a) to X₁+k = x₁ + fti+k f(t, x(t))dt, to find an approximate
formula for the numerical solution of the IVP (t) = f(t, x(t)), x(t₁) = xo.
(c) Use the IVP (t) = 10 x(t) + 11t-5t²-1, x(0) = 0 to test the validity of the formula on the
interval [0, 2] with h = 0.25. [Hint: the exact solution is x(t) = (2) t²-t.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fdf193642-5b60-45f4-a876-cff5951ce07e%2Fd881e496-5557-47a5-a1d1-4d2a16643e2b%2F8p9fidq_processed.jpeg&w=3840&q=75)
Transcribed Image Text:-4h
(a) Derive the Gaussian quadrature formula of the form fh f(x) dx = a f(h) + bf (2h) + cf (3h). Find
the order of convergence of this formula.
(b) Apply the formula you found on part (a) to X₁+k = x₁ + fti+k f(t, x(t))dt, to find an approximate
formula for the numerical solution of the IVP (t) = f(t, x(t)), x(t₁) = xo.
(c) Use the IVP (t) = 10 x(t) + 11t-5t²-1, x(0) = 0 to test the validity of the formula on the
interval [0, 2] with h = 0.25. [Hint: the exact solution is x(t) = (2) t²-t.
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