A software package that supports the construction of

Problem 2RP In Problems 1–4, find a general solution x(t), y(t) for the given system.

Problem 1E A software package that supports the construction of Poincar? maps is required for the problems in this section. Compute and graph the points of the Poincaré map with t = 2?n,n = 0, 1, . . . , 20 for equation (1), taking A = F = 1, ? = 0, and ? = 3/2. Repeat, taking ? = 3/5. Do you think the equation has a -periodic solution for either choice of ?? A sub harmonic solution?

Problem 1RP In Problems 1–4, find a general solution x(t), y(t) for the given system.

In the coupled mass–spring system depicted in Figure 5.26, page 285, take each mass to be 1 kg and let \(k_{1}=8 \mathrm{~N} / \mathrm{m}\), while \(k_{2}=3 \mathrm{~N} / \mathrm{m}\). What are the natural angular frequencies of the system? What is the general solution? Equation transcription: Text transcription: k{1}=8{~N} /{m} k{2}=3{~N} /{m}

A software package that supports the construction of Poincaré maps is required for the problems in this section. Compute and graph the points of the Poincaré map with \(t=2 \pi n, n=0.1, \ldots, 20\) for equation (4), taking \(A=F-1, \varphi=0, \omega=1\), and \(b=0.1\). What is happening to these points as \(n \rightarrow \infty\) ? Equation transcription: Text transcription: (t=2 pi n, n=0.1, ldots, 20\) A=F-1, varphi=0, omega=1 b=0.1 n rightarrow infty

Problem 2E A software package that supports the construction of Poincar? maps is required for the problems in this section. Compute and graph the points of the Poincaré map with t = 2?n,n = 0, 1, . . . , 20 for equation (1), taking A = F = 1, ? = 0, and ? = 1/ . Describe the limit set for this system.

Problem 4E A software package that supports the construction of Poincar? maps is required for the problems in this section. Compute and graph the Poincaré map with t = 2?n,n = 0, 1, . . . , 20 for equation (4), taking A = F = 1, ? = 0, and ? = 1,and b = 0.1. Describe the attractor for this system.

Problem 4RP In Problems 1–4, find a general solution x(t), y(t) for the given system.

In Problems 1–4, find a general solution x(t) , y(t) for the given system. \(2 x^{\prime}-y^{\prime}=y+3 x e^{r}\) , \(3 y^{\prime}-4 x^{\prime}=y-15 x+e^{-t}\) Equation transcription: Text transcription: 2 x^{prime}-y^{prime}=y+3 x e^{r} 3 y^{prime}-4 x^{prime}=y-15 x+e^{-t}

In Problems 5 and 6, solve the given initial value problem. \(x^{\prime}=z-y ; x(0)=0\), \(y^{\prime}=z ; y(0)=0\), \(z^{\prime}=z-x ; z(0)=2\) Equation transcription: Text transcription: x^{prime}=z-y ; x(0)=0 y^{prime}=z ; y(0)=0 z^{prime}=z-x ; z(0)=2

For the interconnected tanks problem of Section , page 242, suppose that instead of pure water being fed into tank , a brine solution with concentration is used; all other data remain the same. Determine the mass of salt in each tank at time if the initial masses are \(x_{0}=0.1 \mathrm{~kg}\) and \(y_{0}=0.3 \mathrm{~kg}\). In Problems , write the given higher-order equation or system in an equivalent normal form (compare Section 5.3). Equation transcription: Text transcription: x{0}=0.1{~kg} y{0}=0.3{~kg}

In Problems 5 and 6, solve the given initial value problem. \(x^{\prime}=y+z ; x(0)=2\), \(y^{\prime}=x+z ; y(0)=2\) \(z^{\prime}=x+y ; z(0)=1\) Equation transcription: Text transcription: x^{prime}=y+z ; x(0)=2 y^{prime}=x+z ; y(0)=2 z^{prime}=x+y ; z(0)=1

In Problems 8–11, write the given higher-order equation or system in an equivalent normal form (compare Section 5.3). \(2 y^{\prime \prime}-t y^{\prime}+8 y=\sin t\) Equation Transcription: Text Transcription: 2y''-ty'+8y=sin? t

In Problems 8–11, write the given higher-order equation or system in an equivalent normal form (compare Section 5.3). \(x^{n}-x+y=0\), Equation transcription: Text transcription: x^{n}-x+y=0

Problem 9RP In Problems 8–11, write the given higher-order equation or system in an equivalent normal form (compare Section 5.3).

Problem 5E A software package that supports the construction of Poincare maps is required for the problems in this section. Compute and graph the Poincaré map with t = 2?n, n = 0, 1, . . . , 20 for equation (4), taking A = F = 1, ? = 0, and ? = 1/3,and b = 022. Describe the attractor for this system. Equation (4)

Problem 11E To show that the limit set for the Poincaré map Xn:= x(2?n), vn:= x’(2?n),where x(t) is a solution to equation (6), is an ellipse and that this ellipse is the same for any initial values x0,v0 , do the following: (a) Argue that since the initial values affect only the transient solution to (6), the limit set for the Poincaré map is independent of the initial values. (b) Now show that for n large, Where And (c) Use the result of part (b) to conclude that the Ellipse contains the limit set of the Poincaré map.

Problem 10RP In Problems 8–11, write the given higher-order equation or system in an equivalent normal form (compare Section 5.3).

Problem 12RP In Problems 12 and 13, solve the phase plane equation for the given system. Then sketch by hand several representative trajectories (with their flow arrows) and describe the stability of the critical points (i.e., compare with Figure 5.12, page 268). Figure 5.12 Examples of different trajectory behaviors near critical point at origin In Problems 12 and 13, solve the phase plane equation for the given system. Then sketch by hand several representative trajectories (with their flow arrows) and describe the stability of the critical points (i.e., compare with Figure 5.12, page 268). Figure 5.12 Examples of different trajectory behaviors near critical point at origin

In Problems 12 and 13, solve the phase plane equation for the given system. Then sketch by hand several representative trajectories (with their flow arrows) and describe the stability of the critical points (i.e., compare with Figure 5.12, page 268). \(x^{\prime}=4-4 y\), \(y^{\prime}=-4 x\) Equation transcription: Text transcription: x^{\prime}=4-4 y y^{\prime}=-4 x

Find all the critical points and determine the phase plane solution curves for the system \(\frac{\dot{d}}{d t}=\sin x \cos y \frac{\dot{w}}{\dot{a}}=\cos x \sin y\) Equation Transcription: Text Transcription: dx/dt=sin? x cos ?y dy/dt=cos? x sin ?y

In Problems 12 and 13, solve the phase plane equation for the given system. Then sketch by hand several representative trajectories (with their flow arrows) and describe the stability of the critical points (i.e., compare with Figure 5.12, page 268). \(x^{\prime}=y-2\), \(y^{\prime}=2-x\) Equation transcription: Text transcription: x^{\prime}=y-2 y^{\prime}=2-x

Problem 11RP In Problems 8–11, write the given higher-order equation or system in an equivalent normal form (compare Section 5.3).

In Problems 15 and 16, sketch some typical trajectories for the given system, and by comparing with Figure 5.12, page 268 identify the type of critical point at the origin. \(x^{\prime}=-2 x-y\), \(y^{\prime}=3 x-y\) Equation transcription: Text transcription: x^{prime}=-2 x-y y^{prime}=3 x-y

Problem 16RP In Problems 15 and 16, sketch some typical trajectories for the given system, and by comparing with Figure 5.12, page 268 identify the type of critical point at the origin. Figure 5.12 Examples of different trajectory behaviors near critical point at origin

Problem 17RP In the electrical circuit of Figure 5.52, take R1=R2=1? , C =1 F, and L =1 H. Derive three equations for the unknown currents I1, I2, and I3 by writing Kirchhoff’s voltage law for loops 1 and 2, and Kirchhoff’s current law for the top juncture. Find the general solution. Figure 5.52 Electrical circuit for Problem 17