Log In Start studying!

Select your language

Suggested languages for you:
Answers without the blur. Sign up and see all textbooks for free! Illustration


Fundamentals Of Differential Equations And Boundary Value Problems
Found in: Page 288
Fundamentals Of Differential Equations And Boundary Value Problems

Fundamentals Of Differential Equations And Boundary Value Problems

Book edition 9th
Author(s) R. Kent Nagle, Edward B. Saff, Arthur David Snider
Pages 616 pages
ISBN 9780321977069

Answers without the blur.

Just sign up for free and you're in.


Short Answer

Suppose the displacement functions xt and yt for a coupled mass-spring system (similar to the one discussed in Problem 6) satisfy the initial value problem

x''(t)+5x(t)-2y(t)=0y''(t)+2y(t)-2x(t)=3sin2tx(0)=x'(0)=0y(0)=1, y'(0)=0

Solve for xt and yt

The solution for x(t)=25cost+45sint-25cos6t+65sin6t-sin2t and


See the step by step solution

Step by Step Solution

Using the elimination procedure

Given system can be written in the following form:


By L1=D2+5,L2=-2,L3=-2,L4=D2+2,f1=0,f2=sin2t, using the elimination procedure,

One obtains L1L4-L2L3[x]=L4f1-L2f2



Finding derivatives

The auxiliary equation is r2+1r2+6=0 and its roots r1,2=±i, r3,4=±i6.

Therefore, the solution to the corresponding homogeneous equation is


Let's check if x(t)=-sin2t is a particular solution for (1).

First, one will find derivatives


Finding x



xp(t)=-sin2t is a particular solution to 1 and the general solution is

x(t)=xh(t)+xp(t) =c1cost+c2sint+c3cos6t+c4sin6t-sin2t .

From the first equation of the system, x''+5x-2y=0 one can find a general solution for yt but the first one needs to find x''



Then simplify,

2y=x''+5x=-c1cost-c2sint-6c3cos6t-6c4sin6t+4sin2t+5c1cost+5c2sint +5c3cos6t+5c4sin6t-5sin2t=4c1cost+4c2sint-c3cos6t-c4sin6t-sin2t

Therefore, y(t)=2c1cost+2c2sint-c32cos6t-c42sin6t-12sin2t

It remains to find constants from initial conditions.

Finding  xt,yt

Let's compute y'



By solving this system, one obtains c1=25, c2=45, c3=-25, c4=65

Finally, x(t)=25cost+45sint-25cos6t+65sin6t-sin2t

And y(t)=45cost+85sint+15cos6t-610sin6t-12sin2t

Most popular questions for Math Textbooks

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.


Want to see more solutions like these?

Sign up for free to discover our expert answers
Get Started - It’s free

Recommended explanations on Math Textbooks

94% of StudySmarter users get better grades.

Sign up for free
94% of StudySmarter users get better grades.