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9. The electric circuit shown in Figure 7.9.1 is described by the system of differential equations \begin{equation} \frac{dx}{dt} = \begin{pmatrix} -\frac{1}{2} & -\frac{1}{8} \\ 2 & -\frac{1}{2} \end{pmatrix} x + \begin{pmatrix} \frac{1}{2} \\ 0 \end{pmatrix} I(t), \end{equation} (49) where $x_1$ is the current through the inductor, $x_2$ is the voltage drop across the capacitor, and $I(t)$ is the current supplied by the external source. a. Determine a fundamental matrix $\Psi(t)$ for the homogeneous system corresponding to equation (49). Refer to Problem 20 of Section 7.6. b. If $I(t) = e^{-t/2}$, determine the solution of the system (49) that also satisfies the initial conditions $x(0) = 0$. $L = 8$ henrys $I(t)$ $C = \frac{1}{2}$ farad $R = 4$ ohms $R = 4$ ohms

          9. The electric circuit shown in Figure 7.9.1 is described by the
system of differential equations
\begin{equation}
\frac{dx}{dt} = \begin{pmatrix} -\frac{1}{2} & -\frac{1}{8} \\ 2 & -\frac{1}{2} \end{pmatrix} x + \begin{pmatrix} \frac{1}{2} \\ 0 \end{pmatrix} I(t),
\end{equation} (49)
where $x_1$ is the current through the inductor, $x_2$ is the voltage drop
across the capacitor, and $I(t)$ is the current supplied by the external
source.
a. Determine a fundamental matrix $\Psi(t)$ for the homogeneous
system corresponding to equation (49). Refer to Problem 20 of
Section 7.6.
b. If $I(t) = e^{-t/2}$, determine the solution of the system (49)
that also satisfies the initial conditions $x(0) = 0$.
$L = 8$ henrys
$I(t)$
$C = \frac{1}{2}$ farad
$R = 4$ ohms
$R = 4$ ohms

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9. The electric circuit shown in Figure 7.9.1 is described by the
system of differential equations

(dx)/(dt) =
< p m a t r i x >
x +
< p m a t r i x >
I(t),
(49)
where x1 is the current through the inductor, x2 is the voltage drop
across the capacitor, and I(t) is the current supplied by the external
source.
a. Determine a fundamental matrix Ψ(t) for the homogeneous
system corresponding to equation (49). Refer to Problem 20 of
Section 7.6.
b. If I(t) = e^-t/2, determine the solution of the system (49)
that also satisfies the initial conditions x(0) = 0.
L = 8 henrys
I(t)
C = (1)/(2) farad
R = 4 ohms
R = 4 ohms

Added by Julio J.

University Physics with Modern Physics

Hugh D. Young 14th Edition

Instant Answer

Solved by Expert Shu-Ting Huang

01/22/2022

Step 1

Step 1: To determine the fundamental matrix for the homogeneous system corresponding to equation (49), we need to first find the eigenvalues and eigenvectors of the coefficient matrix. Show more…

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9. The electric circuit shown in Figure 7.9.1 is described by the system of differential equations 1 dx dt 2 1 8 1 2 (I(t)) x + R = 4 ohms (2)10). 2 I (t), where x₁ is the current through the inductor, x2 is the voltage drop across the capacitor, and I(t) is the current supplied by the external source. a. Determine a fundamental matrix (t) for the homogeneous system corresponding to equation (49). Refer to Problem 20 of Section 7.6. b. If I(t) = e e-t/2, determine the solution of the system (49) that also satisfies the initial conditions x(0) = 0. (49) L = 8 henrys D C = farad R = 4 ohms The electric circuit shown in Figure 7.9.1 is described by the system of differential equations 1 1 1 8 x+ I(t), 0 2 dx dt 2 (49) 2 where x is the current through the inductor,x is the voltage drop across the capacitor, and I(t) is the current supplied by the external source a.Determine a fundamental matrix t for the homogeneous system corresponding to equation (49.Refer to Problem 20 of Section 7.6. b.If It= e-t/2,determine the solution of the system (49 that also satisfies the initial conditions x0=0 L=8henrys farad 2 R=4 ohms R=4ohms

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