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1) Given the following data, find y'(6) 2 26 X y 3 58 4 112 + 7 466 9 922

1) Given the following data, find y'(6) 2 26 X y 3 58 4 112 + 7 466 9 922

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Solve the question according to my source Numerical Analysis -
Newton's Forward Differences Formula to get the derivative We want to find the derivative of y = f(x) passing through the (n+1) points, at a point nearer to the starting value x = xo. Newton's Forward Differences Formula Y(xo+uh) Yo+uAyo + 2! where u=- dy 1 dx dy dy du 1 dy == - dx du dx h du A d²y 1 = dx² h² And also d³y 1 x-xo h . 2u-1, Ayo+ A²y + 2 = (1/2) - 1/12 = dx2 du dx dx du dx h d'y d (dy du d (dy dx³ h³ 24 The above equation gives the value of of general x which may be anywhere in the interval E = 7/34³% + - = u(u-1) u(u-1)(u-2) -A²yo + 3! • u du = dy dx h where y= h A²yo+ (u-1)4³yo + d²y = dx² h² Y(x)= y(x+yh) = Yn+yVy(n-1) + x-xn dy dy dy 1 dy dx dy dx hdy d³y dx3 3 3u²-6u+2 6 12u - 18 12 Newton's backward differences interpolation formula is :- y(y + 1) p²y (n) 2! Newton's Backward Differences Formula to compute the derivative yn + 2y 1, 1/2 √y₁ + ²y + ¹ v²y₂ + 3y² + 6y + 2 Vyn 2 6 -A*yo + -1/2 [0²/₁ + (y + 1) 0³ y + 1 12y + 18 12 -4³% + 6u²-18u+11 12 4u³-18u² +22u-6 -4³% +... 6y² + 18y + 11 12 -Vªyn + ... -Ayo +... -A¹yo + 4y³ + 18y² + 22y +6, +- 24 y(y + 1)(y + 2)p³y(n) +- 3! *y+... vyn+
Transcribed Image Text:Newton's Forward Differences Formula to get the derivative We want to find the derivative of y = f(x) passing through the (n+1) points, at a point nearer to the starting value x = xo. Newton's Forward Differences Formula Y(xo+uh) Yo+uAyo + 2! where u=- dy 1 dx dy dy du 1 dy == - dx du dx h du A d²y 1 = dx² h² And also d³y 1 x-xo h . 2u-1, Ayo+ A²y + 2 = (1/2) - 1/12 = dx2 du dx dx du dx h d'y d (dy du d (dy dx³ h³ 24 The above equation gives the value of of general x which may be anywhere in the interval E = 7/34³% + - = u(u-1) u(u-1)(u-2) -A²yo + 3! • u du = dy dx h where y= h A²yo+ (u-1)4³yo + d²y = dx² h² Y(x)= y(x+yh) = Yn+yVy(n-1) + x-xn dy dy dy 1 dy dx dy dx hdy d³y dx3 3 3u²-6u+2 6 12u - 18 12 Newton's backward differences interpolation formula is :- y(y + 1) p²y (n) 2! Newton's Backward Differences Formula to compute the derivative yn + 2y 1, 1/2 √y₁ + ²y + ¹ v²y₂ + 3y² + 6y + 2 Vyn 2 6 -A*yo + -1/2 [0²/₁ + (y + 1) 0³ y + 1 12y + 18 12 -4³% + 6u²-18u+11 12 4u³-18u² +22u-6 -4³% +... 6y² + 18y + 11 12 -Vªyn + ... -Ayo +... -A¹yo + 4y³ + 18y² + 22y +6, +- 24 y(y + 1)(y + 2)p³y(n) +- 3! *y+... vyn+
1) Given the following data, find y'(6) 0 2 + 26 X y 4 3 58 4 112 7 466 9 922
Transcribed Image Text:1) Given the following data, find y'(6) 0 2 + 26 X y 4 3 58 4 112 7 466 9 922
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