/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Q10E In Problems 10鈥�13, use the vec... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Q10E (page 259)

In Problems 10鈥�13, use the vectorized Euler method with = 0.25 to find an approximation for the solution to the given initial value problem on the specified interval.

y''+ty'+y=0;y(0)=1,y'(0)=0on[0,1]

Short Answer

Expert verified

y(0.25)=1y(0.5)=0.9375y(0.75)=0.8164y(1)=0.651855

Step by step solution

01

Transform equation

Here h=0.25 0n [0,1]

The equations can be written as;

x1(t)=y(t)x2(t)=x'(t)

The transformation of the equation is;

x'1(t)=x2(t)x'2(t)=-x1-tx2

The initial conditions are;

x1(0)=y1(0)=1=x1,0x2(0)=y'(0)=0=x2,0

02

Apply Euler’s method.

Now,

xn+1=xn+hf(tn,xn)

tn+1=tn+h=0+0.25x1(0.25)=x1,1=1x2(0.25)=x2,1=-0.25

And

tn+1=tn+ht2=0.25+0.25x1(0.5)=x1,2=0.9375x2(0.5)=x2,2=-0.484375

t3=0.5+0.25=0.75x1(0.75)=x1,3=0.816406x2(0.75)=x2,3=-0.658203

t4=0.75+0.25=1x1(1)=x1,4=0.651855x2(1)=x2,4=-0.738891

This is the required result.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

A double pendulum swinging in a vertical plane under the influence of gravity (see Figure5.35) satisfies the system

m1+m2l12胃1''+m2l1l2胃2''+m1+m2l1驳胃1=0m2l22胃2''+m2l1l2胃1''+m2l2驳胃2=0

When 胃1and胃2are small angles. Solve the system when

m1=3kg,m2=2kg,l1=l2=5m,胃1(0)=蟺/6,胃2(0)=胃1'(0)=胃2'(0)=0.

In Problems 3 鈥� 18, use the elimination method to find a general solution for the given linear system, where differentiation is with respect to t.

dxdt+y=t2,-x+dydt=1

Find a general solution xt,ytfor the given system.

x''+x-y''=2e-tx''-x+y''=0

Show that the operator (D-1)(D+2) is the same as the operator D2+D-2.

A Problem of Current Interest. The motion of an ironbar attracted by the magnetic field produced by a parallel current wire and restrained by springs (see Figure 5.17) is governed by the equation\(\frac{{{{\bf{d}}^{\bf{2}}}{\bf{x}}}}{{{\bf{d}}{{\bf{t}}^{\bf{2}}}}}{\bf{ = - x + }}\frac{{\bf{1}}}{{{\bf{\lambda - x}}}}\) for \({\bf{ - }}{{\bf{x}}_{\bf{o}}}{\bf{ < x < \lambda }}\)where the constants \({{\bf{x}}_{\bf{o}}}\) and \({\bf{\lambda }}\) are, respectively, the distances from the bar to the wall and to the wire when thebar is at equilibrium (rest) with the current off.

  1. Setting\({\bf{v = }}\frac{{{\bf{dx}}}}{{{\bf{dt}}}}\), convert the second-order equation to an equivalent first-order system.
  2. Solve the related phase plane differential equation for the system in part (a) and thereby show that its solutions are given by\({\bf{v = \pm }}\sqrt {{\bf{C - }}{{\bf{x}}^{\bf{2}}}{\bf{ - 2ln(\lambda - x)}}} \), where C is a constant.
  3. Show that if \({\bf{\lambda < 2}}\) there are no critical points in the xy-phase plane, whereas if \({\bf{\lambda > 2}}\) there are two critical points. For the latter case, determine these critical points.
  4. Physically, the case \({\bf{\lambda < 2}}\)corresponds to a current so high that the magnetic attraction completely overpowers the spring. To gain insight into this, use software to plot the phase plane diagrams for the system when \({\bf{\lambda = 1}}\) and when\({\bf{\lambda = 3}}\).
  5. From your phase plane diagrams in part (d), describe the possible motions of the bar when \({\bf{\lambda = 1}}\) and when\({\bf{\lambda = 3}}\), under various initial conditions.
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Calculus

Read Explanation

Geometry

Read Explanation

Pure Maths

Read Explanation

Applied Mathematics

Read Explanation

Decision Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.