Suppose that we use the Improved Euler's method to approximate the solution to the differential equation dy = x - 2.5y; da Let f(x, y) = x - 2.5y. We let zo = 0.4 and yo = 8 and pick a step size h = 0.25. The improved Euler method is the the following algorithm. From (an, yn), our approximation to the solution of the differential equation at the n-th stage, we find the next stage by computing the x-step n+1 = n + h, and then k₁, the slope at (an, yn). The predicted new value of the solution is Zn+1 = yn + h.k₁. Then we find the slope at the predicted new point k₂ = f(n+1, Zn+1) and get the corrected point by averaging slopes Complete the following table: n xn Yn k₁ Zn+1 k2 0 0.4 8 -19.6 3.1 -7.1 Yn+1 = Yn + Thus the actual value of the function at the point x = 1.4 is y(1.4)= -0 h 2 y(0.4) = 8. The exact solution can also be found for the linear equation. Write the answer as a function of a. y(x) = (k₁ +k₂).
Suppose that we use the Improved Euler's method to approximate the solution to the differential equation dy = x - 2.5y; da Let f(x, y) = x - 2.5y. We let zo = 0.4 and yo = 8 and pick a step size h = 0.25. The improved Euler method is the the following algorithm. From (an, yn), our approximation to the solution of the differential equation at the n-th stage, we find the next stage by computing the x-step n+1 = n + h, and then k₁, the slope at (an, yn). The predicted new value of the solution is Zn+1 = yn + h.k₁. Then we find the slope at the predicted new point k₂ = f(n+1, Zn+1) and get the corrected point by averaging slopes Complete the following table: n xn Yn k₁ Zn+1 k2 0 0.4 8 -19.6 3.1 -7.1 Yn+1 = Yn + Thus the actual value of the function at the point x = 1.4 is y(1.4)= -0 h 2 y(0.4) = 8. The exact solution can also be found for the linear equation. Write the answer as a function of a. y(x) = (k₁ +k₂).
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:Suppose that we use the Improved Euler's method to approximate the solution to the differential equation
dy
= x - 2.5y;
dx
Let f(x, y) = x - 2.5y.
We let x = 0.4 and yo = 8 and pick a step size h = 0.25. The improved Euler method is the the following algorithm. From (æn, Yn), our approximation to the solution of the
differential equation at the n-th stage, we find the next stage by computing the x-step n+1 = n + h, and then k₁, the slope at (xn, Yn). The predicted new value of the solution is
Zn+1 = Yn+h.k₁. Then we find the slope at the predicted new point k₂ = f(n+1, Zn+1) and get the corrected point by averaging slopes
Complete the following table:
n xn Yn
k₁ Zn+1 k2
0 0.4 8 -19.6 3.1 -7.1
2
3
Yn+1 Yn +
Thus the actual value of the function at the point x = 1.4 is
y(1.4)=
h
2
y(0.4) = 8.
The exact solution can also be found for the linear equation. Write the answer as a function of a.
y(x) =
(k₁ +k₂).
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