The Taylor method of order 2 can be used to approximate the solution to the initial value problem y' = y. y(0)=1 at x = 1. Show that the approximation y obtained by using the Taylor method of order 2, with step size is given by the formula y₁=1+ The solution to the initial value problem is y=e*, so yn is an approximation of the constante. Let y'=f(x,y) and y'=f₂(x,y). If xn+1 = x +h, then which recursive formula, for Yn+1. represents the Taylor method of order 2? O A. Yn+1=Yn + f(xn-Yn) + 12 (xn-Yn) • Yn+1 = Yn +hf (x + 2₁ Yn + ²(xn³Yn)] OC. Yn+1+Yn+hf(xn+h. Yn+hf(xn-Yn)) OB. Yn+h h² OD. Yn+1=Yn+hf (XnYn) + 21¹2(Xn-Yn) In this case, what is the value of h? h=0 If y'=y, what can be said about each successive derivative of y? OA. Each successive derivative of y is equal to some constant C. OB. Each successive derivative of y is equal to 0. OC. Each successive derivative of y is equal to y plus some constant C OD. Each successive derivative of y is equal to y. Using the information in the previous step, find f₂(x,y). f₂(x,y) = Substitute the values of h, f(xnYn), and f₂ (XnYn) into the formula for Yn+1 Yn+1=0 for n = 1, 2
The Taylor method of order 2 can be used to approximate the solution to the initial value problem y' = y. y(0)=1 at x = 1. Show that the approximation y obtained by using the Taylor method of order 2, with step size is given by the formula y₁=1+ The solution to the initial value problem is y=e*, so yn is an approximation of the constante. Let y'=f(x,y) and y'=f₂(x,y). If xn+1 = x +h, then which recursive formula, for Yn+1. represents the Taylor method of order 2? O A. Yn+1=Yn + f(xn-Yn) + 12 (xn-Yn) • Yn+1 = Yn +hf (x + 2₁ Yn + ²(xn³Yn)] OC. Yn+1+Yn+hf(xn+h. Yn+hf(xn-Yn)) OB. Yn+h h² OD. Yn+1=Yn+hf (XnYn) + 21¹2(Xn-Yn) In this case, what is the value of h? h=0 If y'=y, what can be said about each successive derivative of y? OA. Each successive derivative of y is equal to some constant C. OB. Each successive derivative of y is equal to 0. OC. Each successive derivative of y is equal to y plus some constant C OD. Each successive derivative of y is equal to y. Using the information in the previous step, find f₂(x,y). f₂(x,y) = Substitute the values of h, f(xnYn), and f₂ (XnYn) into the formula for Yn+1 Yn+1=0 for n = 1, 2
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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Transcribed Image Text:1
is given by the formula y =
The Taylor method of order 2 can be used to approximate the solution to the initial value problem y' = y, y(0) = 1 at x = 1. Show that the approximation y obtained by using the Taylor method of order 2, with step size
n
The solution to the initial value problem is y=e*, so yn is an approximation of the constante.
Let y'=f(x,y) and y'' = f₂(x,y). If Xn+1 = X₁ +h, then which recursive formula, for Yn+1, represents the Taylor method of order 2?
h
O A. Yn+1=Yn + f(xn Yn) + ¹₂ (xn-Yn)
h
h
• + h²{x₁ + 1 = ₁ yn + ·
+hf/xn- in + 1/{f(xnxn))
OC. Yn+1+Yn +hf(xn+h, Yn+hf(xn Yn))
O B. Yn+1=Yn+h
h²
OD. Yn+1=Yn +hf(xn-Yn) + 21¹2(XnYn)
In this case, what is the value of h?
h=
If y' = y, what can be said about each successive derivative of y?
O A. Each successive derivative of y is equal to some constant C.
O B. Each successive derivative of y is equal to 0.
O C. Each successive derivative of y is equal to y plus some constant C.
O D. Each successive derivative of y is equal to y.
Using the information in the previous step, find f₂(x,y).
f₂ (x,y) =
Substitute the values of h, f(xn Yn), and f₂ (XnYn) into the formula for Yn+1
Yn+1 =
C
n
1 1
«=[+++ 2)^•
1+
n 2n
, for n = 1, 2,...
Transcribed Image Text:Why is the equation in the problem statement true for n = 1, 2,...?
O A. Raising both sides of the equation in the previous step to the power of n, results in the desired equation.
O B. Substituting n = 1, 2,... into the equation in the previous step always results in a true equation.
times the previous term yn
OC. Each term yn +1 is always 1+
+
n
2n²
O D. Since the differential equation y'=y is true, the corresponding difference equation Yn+1 = Yn is also true. Making the substitution Yn+1 = Yn results in the desired equation.
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