Solution: In Exercises 3–6, solve the initial value problem

Problem 22E [M] The circuit in the figure is described by the equation where iL is the current through the inductor L and vC is the voltage drop across the capacitor C. Find formulas for iL and vC when R = .5 ohm, C = 2.5 farads, L = .5 henry, the initial current is 0 amp, and the initial voltage is 12 volts.

Problem 1E A particle moving in a planar force field has a position vector x that satisfies matrix A has eigenvalues 4 and 2, with corresponding eigenvectors Find the position of the particle at time t, assuming that

Problem 2E

In Exercises 3–6, solve the initial value problem \(\mathbf{x}^{\prime}(t)=A \mathbf{x}(t)\) for \(t \geq 0\), with \(\mathbf{x}(0)=(3,2)\). Classify the nature of the origin as an attractor, repeller, or saddle point of the dynamical system described by \(\mathbf{x}^{\prime}=A \mathbf{x}\). Find the directions of greatest attraction and/or repulsion. When the origin is a saddle point, sketch typical trajectories. \(A=\left[\begin{array}{rr}2 & 3 \\ -1 & -2\end{array}\right]\)

Problem 4E In Exercises 3–6, solve the initial value problem for . Classify the nature of the origin as an attractor, repeller, or saddle point of the dynamical system described by .Find the directions of greatest attraction and/or repulsion. When the origin is a saddle point, sketch typical trajectories.

In Exercises 3–6, solve the initial value problem \(\mathbf{x}^{\prime}(t)=A \mathbf{x}(t)\) for \(t \geq 0\), with \(\mathbf{x}(0)=(3,2)\). Classify the nature of the origin as an attractor, repeller, or saddle point of the dynamical system described by \(\mathbf{x}^{\prime}=A \mathbf{x}\). Find the directions of greatest attraction and/or repulsion. When the origin is a saddle point, sketch typical trajectories. \(A=\left[\begin{array}{rr}7 & -1 \\ 3 & 3\end{array}\right]\)

Problem 7E In Exercises 7 and 8, make a change of variable that decouples the equation . Write the equation and show the calculation that leads to the uncoupled system specifying P and D. A as in Exercise 5 Reference: In Exercises 3–6, solve the initial value problem for . Classify the nature of the origin as an attractor, repeller, or saddle point of the dynamical system described by .Find the directions of greatest attraction and/or repulsion. When the origin is a saddle point, sketch typical trajectories.

Problem 6E In Exercises 3–6, solve the initial value problem for . Classify the nature of the origin as an attractor, repeller, or saddle point of the dynamical system described by .Find the directions of greatest attraction and/or repulsion. When the origin is a saddle point, sketch typical trajectories.

Problem 9E In Exercises 9–18, construct the general solution of involving complex eigenfunctions and then obtain the general real solution. Describe the shapes of typical trajectories.

Problem 8E In Exercises 7 and 8, make a change of variable that decouples the equation . Write the equation and show the calculation that leads to the uncoupled system specifying P and D. A as in Exercise 6 Reference: In Exercises 3–6, solve the initial value problem for . Classify the nature of the origin as an attractor, repeller, or saddle point of the dynamical system described by .Find the directions of greatest attraction and/or repulsion. When the origin is a saddle point, sketch typical trajectories.

In Exercises 9–18, construct the general solution of \(\mathbf{x}^{\prime}=A \mathbf{x}\) involving complex eigenfunctions and then obtain the general real solution. Describe the shapes of typical trajectories. \(A=\left[\begin{array}{rr}3 & 1 \\ -2 & 1\end{array}\right]\)

Problem 12E In Exercises 9–18, construct the general solution of involving complex eigenfunctions and then obtain the general real solution. Describe the shapes of typical trajectories.

In Exercises 9–18, construct the general solution of \(\mathbf{x}^{\prime}=A \mathbf{x}\) involving complex eigenfunctions and then obtain the general real solution. Describe the shapes of typical trajectories. \(A=\left[\begin{array}{rr}-3 & -9 \\ 2 & 3\end{array}\right]\)

In Exercises 9–18, construct the general solution of \(\mathbf{x}^{\prime}=A \mathbf{x}\) involving complex eigenfunctions and then obtain the general real solution. Describe the shapes of typical trajectories. \(A=\left[\begin{array}{ll}4 & -3 \\ 6 & -2\end{array}\right]\)

Problem 14E In Exercises 9–18, construct the general solution of involving complex eigenfunctions and then obtain the general real solution. Describe the shapes of typical trajectories.

Problem 15E In Exercises 9–18, construct the general solution of involving complex eigenfunctions and then obtain the general real solution. Describe the shapes of typical trajectories.

Problem 16E In Exercises 9–18, construct the general solution of involving complex eigenfunctions and then obtain the general real solution. Describe the shapes of typical trajectories.

In Exercises 9–18, construct the general solution of \(\mathbf{x}^{\prime}=A \mathbf{x}\) involving complex eigenfunctions and then obtain the general real solution. Describe the shapes of typical trajectories. [M] \(A=\left[\begin{array}{rrr}53 & -30 & -2 \\ 90 & -52 & -3 \\ 20 & -10 & 2\end{array}\right]\)

In Exercises 9–18, construct the general solution of \(\mathbf{x}^{\prime}=A \mathbf{x}\) involving complex eigenfunctions and then obtain the general real solution. Describe the shapes of typical trajectories. \([\mathbf{M}] A=\left[\begin{array}{rrr}30 & 64 & 23 \\ -11 & -23 & -9 \\ 6 & 15 & 4\end{array}\right]\)

Problem 19E [M] Find formulas for the voltages v1 and v2 (as functions of time t ) for the circuit in Example 1, assuming that farads, and the initial charge on each capacitor is 4 volts. Reference:

Problem 20E [M] Find formulas for the voltages v1 and v2 for the circuit in Example 1, assuming that farads, and the initial charge on each capacitor is 3 volts. Reference:

Problem 21E [M] Find formulas for the current iL and the voltage vC for the circuit in Example 3, assuming that henry, the initial current is 0 amp, and the initial voltage is 15 volts. Reference: