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Consider the initial value problem given below. y'= -5Y₁ y(0)=9 Show that when Euler's method is used to approximate the solution to this initial value problem at x = 4, then the approximation with step size h is 91- Apply one iteration of Euler's method in order to write the next term yn+1 in terms of xn. Yn and h, as needed. Yn+1 = Substitute in f(xn-Yn) = -5½n into the previous form. Yn+1= Use this recursive relationship to write a term k steps in the future Ynk in terms of xn. Yn. h, and k, as needed. Yn+k= For the approximation in question, write the number of steps that are needed in terms of the step size h. Thus, the approximation k = steps after y₁ = is 9(1 1-

Consider the initial value problem given below. y'= -5Y₁ y(0)=9 Show that when Euler's method is used to approximate the solution to this initial value problem at x = 4, then the approximation with step size h is 91- Apply one iteration of Euler's method in order to write the next term yn+1 in terms of xn. Yn and h, as needed. Yn+1 = Substitute in f(xn-Yn) = -5½n into the previous form. Yn+1= Use this recursive relationship to write a term k steps in the future Ynk in terms of xn. Yn. h, and k, as needed. Yn+k= For the approximation in question, write the number of steps that are needed in terms of the step size h. Thus, the approximation k = steps after y₁ = is 9(1 1-

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter5: Inverse, Exponential, And Logarithmic Functions
Section: Chapter Questions
Problem 18T
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Consider the initial value problem given below. y' = - y(0) = 9 Show that when Euler's method is used to approximate the solution to this initial value problem at x = 4, then the approximation with step size h is 9 (₁-1) ' Apply one iteration of Euler's method in order to write the next term yn + 1 in terms of xn. Yn, and h, as needed. Yn+1 = 1 Substitute in f(xn Yn) = -5Yn into the previous form. Yn+1 = Use this recursive relationship to write a term k steps in the future Yn+k in terms of Xn: Yn, h, and k, as needed. Yn+k= For the approximation in question, write the number of steps that are needed in terms of the step size h. k= = is 9 (1-1/2) a Thus, the approximation k= steps after y₁=
Transcribed Image Text:Consider the initial value problem given below. y' = - y(0) = 9 Show that when Euler's method is used to approximate the solution to this initial value problem at x = 4, then the approximation with step size h is 9 (₁-1) ' Apply one iteration of Euler's method in order to write the next term yn + 1 in terms of xn. Yn, and h, as needed. Yn+1 = 1 Substitute in f(xn Yn) = -5Yn into the previous form. Yn+1 = Use this recursive relationship to write a term k steps in the future Yn+k in terms of Xn: Yn, h, and k, as needed. Yn+k= For the approximation in question, write the number of steps that are needed in terms of the step size h. k= = is 9 (1-1/2) a Thus, the approximation k= steps after y₁=
Expert Solution
Step 1

It is given that y'=-15y with y0=9.

The objective is to show that when Euler method is used to approximate the initial value problem at x=4, then the approximation with step size h is 91-h54h.

The Euler method is a numerical method to obtain the approximate solution.

It starts with the known value yx0=y0 and computes the unknown values y1,y2,⋯,yn successively using the formula yi+1=yi+hfxi,yi, where y'=fx,y is the given differential equation.

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