Estimate the differential equation: y' = y – x – 1, y(0) = 1 Estimate y(1) using a step size h = 0.5 a) Use Euler's Method to estimate y(1) Euler's Method Yn+1 = Yn + hf(xn, Yn) |f(Xn, Ka) Yn+1 1 0.5 0.5 0.5 Improved Euler's Method Yn+1 = Yn +F (xn, Yn) + f(xn+1+Yn+1) Yn+1 = Yn + hf (xn, Yn) b) Use Improved Euler's Method to estimate y(1) h f(Xa. Ka) |f(Xa. Ka") Yn+1 1 0.5 0.5 0.5
Estimate the differential equation: y' = y – x – 1, y(0) = 1 Estimate y(1) using a step size h = 0.5 a) Use Euler's Method to estimate y(1) Euler's Method Yn+1 = Yn + hf(xn, Yn) |f(Xn, Ka) Yn+1 1 0.5 0.5 0.5 Improved Euler's Method Yn+1 = Yn +F (xn, Yn) + f(xn+1+Yn+1) Yn+1 = Yn + hf (xn, Yn) b) Use Improved Euler's Method to estimate y(1) h f(Xa. Ka) |f(Xa. Ka") Yn+1 1 0.5 0.5 0.5
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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Equations and Inequations
Equations and inequalities describe the relationship between two mathematical expressions.
Linear Functions
A linear function can just be a constant, or it can be the constant multiplied with the variable like x or y. If the variables are of the form, x2, x1/2 or y2 it is not linear. The exponent over the variables should always be 1.
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Transcribed Image Text:c) Use Runge-Kutta (RK4) Method to estimate y(1)
umwm
h
k2
k3
k4
Уn+1
1
0.5
0.5
0.5
1
Runge-Kutta
Yn+1 = Yn +(k, + 2k2 + 2k3 + k4)
k, = f(xp, Yn)
k2 = f(xn + » Yn + 쓸)
k3 = f(xn +5 Yn +)
k4 = f (Xn + h, Yn + k3)
h
Transcribed Image Text:Estimate the differential equation:
у' %3Dу — х—1, у(0) 3D1
Estimate y(1) using a step size h = 0.5
a) Use Euler's Method to estimate y(1)
Euler's Method
Yn+1 = Yn + hf (xn, Yn)
h
|f(Xa, Ka)
Yn+1
1
0.5
0.5
0.5
Improved Euler's Method
b) Use Improved Euler's Method to estimate y(1)
Yn+1 = Yn +
Yn+1 = Yn + hf (xn, Yn)
h
f (Xa, Ka) Xa
|f(Xa, Ka")
Уn+1
1
0.5
0.5
0.5
1
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