Euler's method for a first order IVP y' = f(x, y), y(x₁) yo is the the following algorithm. From (xo, yo) we define a sequence of approximations to the solution of the differential equation so that at the nth stage, we have = Xn = xn−1+h, Yn = Yn−1+h⋅ f(xn−1, Yn-1). In this exercise we consider the IVP y' = = -x + y with y(4) = -2. This equation is first order linear with exact solution y = 1 + x - 7eª-4. Use Euler's method with h = 0.1 to approximate the solution of the differential equation. For this example we include the slope field to give a rough idea what the shape of the solution should look like. We have also plotted the exact solution y = 1 + x - 7e²-4 over a small interval. Apply Euler's method to complete the following table: In the first two rows enter the values of an and yn and in the third row use the exact solution to find the errors en = |y(xn) — Yn|. A calculator or other scientific software would be handy to work these types of problem. You can always use answers given by explicit formulas which are very accurate. You need at least 4 significant digits. If your answer is marked wrong try entering a more accurate answer.
Euler's method for a first order IVP y' = f(x, y), y(x₁) yo is the the following algorithm. From (xo, yo) we define a sequence of approximations to the solution of the differential equation so that at the nth stage, we have = Xn = xn−1+h, Yn = Yn−1+h⋅ f(xn−1, Yn-1). In this exercise we consider the IVP y' = = -x + y with y(4) = -2. This equation is first order linear with exact solution y = 1 + x - 7eª-4. Use Euler's method with h = 0.1 to approximate the solution of the differential equation. For this example we include the slope field to give a rough idea what the shape of the solution should look like. We have also plotted the exact solution y = 1 + x - 7e²-4 over a small interval. Apply Euler's method to complete the following table: In the first two rows enter the values of an and yn and in the third row use the exact solution to find the errors en = |y(xn) — Yn|. A calculator or other scientific software would be handy to work these types of problem. You can always use answers given by explicit formulas which are very accurate. You need at least 4 significant digits. If your answer is marked wrong try entering a more accurate answer.
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:Euler's method for a first order IVP y' = f(x, y), y(x) = yo is the the following algorithm.
From (xo, yo) we define a sequence of approximations to the solution of the differential equation
so that at the nth stage, we have
In this exercise we consider the IVP y' = -x + y with y(4)
linear with exact solution y 1+x - 7e²-4
=
Use Euler's method with h
=
Xn = Xn−1+h, Yn =
n = 0
Xn 4 4.1
Yn -2 -4.9
en
For this example we include the slope field to give a rough idea what the shape of the solution
should look like. We have also plotted the exact solution y : 1 + x - 7e-4 over a small
interval.
=
0
Yn-1 + h・ f(xn-1, Yn-1).
Apply Euler's method to complete the following table:
In the first two rows enter the values of în and yn and in the third row use the exact solution to
find the errors en = |y(xn) — Yn |. A calculator or other scientific software would be handy to
work these types of problem. You can always use answers given by explicit formulas which are
very accurate. You need at least 4 significant digits. If your answer is marked wrong try entering a
more accurate answer.
1
=
0.1 to approximate the solution of the differential equation.
4.2
-2. This equation is first order
2
4.3
3
4.4
4
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