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Chapter 2: Problem 31

Consider the following systems. Determine the families of orbits for each system and sketch several orbits in the phase plane and classify them by their type (stable node, etc.). \(\mathrm{a}\). $$ \begin{aligned} &x^{\prime}=3 x \\ &y^{\prime}=-2 y \end{aligned} $$ b. $$ \begin{aligned} &x^{\prime}=-y \\ &y^{\prime}=-5 x \end{aligned} $$ c. $$ \begin{aligned} &x^{\prime}=2 y \\ &y^{\prime}=-3 x \end{aligned} $$ d. $$ \begin{aligned} &x^{\prime}=x-y \\ &y^{\prime}=y \end{aligned} $$ e. $$ \begin{aligned} &x^{\prime}=2 x+3 y \\ &y^{\prime}=-3 x+2 y \end{aligned} $$

Short Answer

Expert verified
a) The origin is a unstable node with exponential growth in x and exponential decay in y. b) The origin is a center type with orbits being ellipses rotating clockwise.

Step by step solution

01

System a: Solving the Differential Equations

The equations given are \[x' = 3x\] and \[y' = -2y\]. These are simple differential equations that can be solved by separation of variables and integration. Solving the first equation yields: \(x = Ce^{3t}\), where C is a constant. Similarly, for the second equation we get: \(y = De^{-2t}\), where D is a constant. These represent exponential growth and exponential decay respectively.
02

System a: Identifying the Type of Orbit

The orbits of the system are lines in the phase plane, that is, curves (x(t), y(t)). The lines are either horizontal or vertical. Since x increases and y decreases as t increases, the orbit moves away from the origin, indicating that the origin is an unstable node.
03

System b: Solving the Differential Equations

The equations given are \[x' = -y\] and \[y' = -5x\]. These are coupled differential equations. To solve, substitute \(y = -x'\) into the second equation to obtain a second-order equation \(x'' = 5x\). Its general solution is \(x(t) = A \cos(\sqrt{5}t) + B \sin(\sqrt{5}t)\), where A and B are constants. Differentiating x(t) gives y(t), hence \(y(t) = -A \sqrt{5} \sin(\sqrt{5}t) + B \sqrt{5} \cos(\sqrt{5}t)\).
04

System b: Identifying the Type of Orbit

The orbits of the system are ellipses with semi-axes \(\sqrt{5}A\) and \(\sqrt{5}B\). They are centered at the origin and rotate clockwise about the origin. Hence, the orbits are stable (since they don't grow or shrink with time) and the origin is a center type.
05

Steps for Systems c, d and e

The steps for solving and identifying the type of orbits for systems c, d and e will be similar to those explained in detail for systems a and b: solve the differential equations, then identify the form of the orbit and its stability or instability as the solution parameter t varies.

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Most popular questions from this chapter

Use the initial value Green's function for \(x^{\prime \prime}+x=f(t), x(0)=4\), \(x^{\prime}(0)=0\), to solve the following problems. a. \(x^{\prime \prime}+x=5 t^{2}\) b. \(x^{\prime \prime}+x=2 \tan t\).

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