please solve 15.2, thanks. Exercise 15.2. Show that the discretized energy defined in (15.5) decreases as n in thediscretized system (15.6).ReferenceExercise 15.1. Show that the discretized energy1en = ;v+(15.5)2increases as n in the discretized system (15.4). 69implicit methodIn order to deal with the deficit of the Euler method in Exercise 15.1, we consider an'implicit' method which approximates the right hand side of (15.3) by the value of newtime step:In+1 - xn= Yn+1,At(15.6)Yn+1 - Yn-w?In+1.Atmass-spring systemThe simplest example (cf. Example 9.1) is an equation describing the motion of amass attached to a spring:ma" = -kx, x(0) = a, x'(0)(15.1)= 6In this system, the total energykmE =(15.2)= meis preserved. The equation (15.1) is reformulated into a system of ODES of the first order:()-(-)-(G)) --d1k(15.3)dt-w2 0mThe first order Euler method (14.4) or (14.5) approximates the system (15.3) byIn+1 - In= Yn,At(15.4)Yn+1YnAtwhere anandYndenote values of x and y att = nAt, respectively.

Name: please solve 15.2, thanks. Exercise 15.2. Show that the discretized en
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Description: please solve 15.2, thanks. Exercise 15.2. Show that the discretized energy defined in (15.5) decreases as n in thediscretized system (15.6).ReferenceExercise 15.1. Show that the discretized energy1en = ;v+(15.5)2increases as n in the discretized syst