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Chapter 5: Q11RP (page 306)

Write the given higher-order equation or system in an equivalent normal form (compare Section \(5.3\)).

\(\begin{array}{l}{\bf{x''' + y' + y'' = t }}\\{\bf{x'' - x' + y''' = 0}}\end{array}\)

Short Answer

Expert verified

The equivalent normal form of the given is:

\(\begin{array}{c}{{\bf{x}}_{\bf{1}}}{\bf{' = }}{{\bf{x}}_{\bf{2}}}{\bf{,}}\;\;\;{{\bf{x}}_{\bf{2}}}{\bf{' = }}{{\bf{x}}_{\bf{3}}}{\bf{,}}\;\;\;{{\bf{x}}_{\bf{3}}}{\bf{' = t - }}{{\bf{x}}_{\bf{5}}}{\bf{ - }}{{\bf{x}}_{\bf{6}}}{\bf{,}}\\{{\bf{x}}_{\bf{4}}}{\bf{' = }}{{\bf{x}}_{\bf{5}}}{\bf{,}}\;\;\;{{\bf{x}}_{\bf{5}}}{\bf{' = }}{{\bf{x}}_{\bf{6}}}{\bf{,}}\;\;\;{{\bf{x}}_{\bf{6}}}{\bf{' = }}{{\bf{x}}_{\bf{2}}}{\bf{ - }}{{\bf{x}}_{\bf{3}}}{\bf{.}}\end{array}\)

Step by step solution

01

Rewrite the given equation

Rewrite the given system as:

\(\begin{array}{c}{\bf{x''' = t - y' - y'' }}\\{\bf{y''' = x' - x''}}\end{array}\)

Setting \({\bf{x = }}{{\bf{x}}_{\bf{1}}}{\bf{,x}}'{\bf{ = }}{{\bf{x}}_{\bf{2}}}{\bf{,x}}''{\bf{ = }}{{\bf{x}}_{\bf{3}}}{\bf{,y = }}{{\bf{x}}_{\bf{4}}}{\bf{,y}}'{\bf{ = }}{{\bf{x}}_{\bf{5}}}\) and \({\bf{y'' = }}{{\bf{x}}_{\bf{6}}}.\)

02

Finding the derivatives of \({\bf{x}}\)

Therefore,the equivalent normal form;

\(\begin{array}{c}{{\bf{x}}_{\bf{1}}}{\bf{' = }}{{\bf{x}}_{\bf{2}}}{\bf{,}}\;\;\;{{\bf{x}}_{\bf{2}}}{\bf{' = }}{{\bf{x}}_{\bf{3}}}{\bf{,}}\;\;\;{{\bf{x}}_{\bf{3}}}{\bf{' = t - }}{{\bf{x}}_{\bf{5}}}{\bf{ - }}{{\bf{x}}_{\bf{6}}}{\bf{,}}\\{{\bf{x}}_{\bf{4}}}{\bf{' = }}{{\bf{x}}_{\bf{5}}}{\bf{,}}\;\;\;{{\bf{x}}_{\bf{5}}}{\bf{' = }}{{\bf{x}}_{\bf{6}}}{\bf{,}}\;\;\;{{\bf{x}}_{\bf{6}}}{\bf{' = }}{{\bf{x}}_{\bf{2}}}{\bf{ - }}{{\bf{x}}_{\bf{3}}}{\bf{.}}\end{array}\)

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Most popular questions from this chapter

Show that the Poincare map for equation (1) is not chaoticby showing that if\({\bf{(}}{{\bf{x}}_{\bf{o}}}{\bf{,}}{{\bf{\nu }}_{\bf{o}}}{\bf{)}}\)and\({\bf{(}}{{\bf{x}}^{\bf{*}}}_{\bf{o}}{\bf{,}}{{\bf{\nu }}^{\bf{*}}}_{\bf{o}}{\bf{)}}\)are two initial values that define the Poincare maps\({\bf{(}}{{\bf{x}}_{\bf{n}}}{\bf{,}}{{\bf{\nu }}_{\bf{n}}}{\bf{)}}\) and\({\bf{(}}{{\bf{x}}^{\bf{*}}}_{\bf{n}}{\bf{,}}{{\bf{\nu }}^{\bf{*}}}_{\bf{n}}{\bf{)}}\), respectively, using the recursive formulas in (3), then one can make the distance between\({\bf{(}}{{\bf{x}}_{\bf{n}}}{\bf{,}}{{\bf{\nu }}_{\bf{n}}}{\bf{)}}\)and\({\bf{(}}{{\bf{x}}^{\bf{*}}}_{\bf{n}}{\bf{,}}{{\bf{\nu }}^{\bf{*}}}_{\bf{n}}{\bf{)}}\)small by making the distance between\({\bf{(}}{{\bf{x}}_{\bf{o}}}{\bf{,}}{{\bf{\nu }}_{\bf{o}}}{\bf{)}}\) and \({\bf{(}}{{\bf{x}}^{\bf{*}}}_{\bf{o}}{\bf{,}}{{\bf{\nu }}^{\bf{*}}}_{\bf{o}}{\bf{)}}\)small. (Hint: Let \({\bf{(A,}}\phi {\bf{)}}\)and \({\bf{(}}{{\bf{A}}^{\bf{*}}}{\bf{,}}{\phi ^ * }{\bf{)}}\) be the polar coordinates of two points in the plane. From the law of cosines, it follows that the distance d between them is given by\({{\bf{d}}^{\bf{2}}}{\bf{ = (A - }}{{\bf{A}}^{\bf{*}}}{{\bf{)}}^{\bf{2}}}{\bf{ + 2A}}{{\bf{A}}^{\bf{*}}}{\bf{(1 - cos(}}\phi {\bf{ - }}{\phi ^ * }{\bf{))}}\).)

The motion of a pair of identical pendulums coupled with a spring is modeled by the system

mx1''=-mglx1-kx1-x2,mx2''=-mglx2+kx1-x2

for small displacements (see Figure 5.36). Determine the two normal frequencies for the system.

Use the result of Problem 31 to prove that all solutions to the equation\({\bf{y'' + }}{{\bf{y}}^{\bf{3}}}{\bf{ = 0}}\)remain bounded. [Hint: Argue that \(\frac{{{{\bf{y}}^{\bf{4}}}}}{{\bf{4}}}\) is bounded above by the constant appearing in Problem 31.]

In Problems 11鈥�14, solve the related phase plane differential equation for the given system. Then sketch by hand several representative trajectories (with their flow arrows).

dxdt=(y-x)(y-1)dydt=(x-y)(x-1)

In Problems 19鈥�24, convert the given second-order equation into a first-order system by setting v=y鈥�. Then find all the critical points in the yv-plane. Finally, sketch (by hand or software) the direction fields, and describe the stability of the critical points (i.e., compare with Figure 5.12).

d2ydt2+y=0
See all solutions

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