91Ó°ÊÓ

The voltage in an \(R L C\) network is described by the differential equation \\[\frac{d^{2} v}{d t^{2}}+4 \frac{d v}{d t}+4 v=0\\] subject to the initial conditions \(v(0)=1\) and \(d v(0) / d t=-1 .\) Determine the characteristic equation. Find \(v(t)\) for \( t> 0\).

The branch current in an \(R L C\) circuit is described by the differential equation \\[\frac{d^{2} i}{d t^{2}}+6 \frac{d i}{d t}+9 i=0\\] and the initial conditions are \(i(0)=0\),\(d i(0) / d t=4 .\) Obtain the characteristic equation and determine \(i(t)\) for \(t > 0\).

The current in an \(R L C\) circuit is described by \\[\frac{d^{2} i}{d t^{2}}+10 \frac{d i}{d t}+25 i=0\\] If \(i(0)=10\) and \(d i(0) / d t=0,\) find \(i(t)\) for \(t > 0\).

The differential equation that describes the voltage in an \(R L C\) network is \\[\frac{d^{2} v}{d t^{2}}+5 \frac{d v}{d t}+4 v=0\\] Given that \(v(0)=0, d v(0) / d t=10,\) obtain \(v(t)\).

The natural response of an \(R L C\) circuit is described by the differential equation \\[\frac{d^{2} v}{d t^{2}}+2 \frac{d v}{d t}+v=0\\] for which the initial conditions are \(v(0)=10\) and \(d v(0) / d t=0 .\) Solve for \(v(t)\).

If \(R=20 \Omega, L=0.6 \mathrm{H},\) what value of \(C\) will make an \(R L C\) series circuit: (a) overdamped, (b) critically damped, (c) underdamped?

The step response of an \(R L C\) circuit is given by \\[\frac{d^{2} i}{d t^{2}}+2 \frac{d i}{d t}+5 i=10\\] Given that \(i(0)=2\) and \(d i(0) / d t=4,\) solve for \(i(t)\).

A branch voltage in an \(R L C\) circuit is described by \\[\frac{d^{2} v}{d t^{2}}+4 \frac{d v}{d t}+8 v=24\\] If the initial conditions are \(v(0)=0=d v(0) / d t\), find \(v(t)\).

The current in an \(R L C\) network is governed by the differential equation \\[\frac{d^{2} i}{d t^{2}}+3 \frac{d i}{d t}+2 i=4\\] subject to \(i(0)=1, d i(0) / d t=-1 .\) Solve for \(i(t)\).

Solve the following differential equations subject to the specified initial conditions (a) \(d^{2} v / d t^{2}+4 v=12, v(0)=0, d v(0) / d t=2\) (b) \(d^{2} i / d t^{2}+5 d i / d t+4 i=8, i(0)=-1\), \(d i(0) / d t=0\) (c) \(d^{2} v / d t^{2}+2 d v / d t+v=3, v(0)=5\), \(d v(0) / d t=1\) (d) \(d^{2} i / d t^{2}+2 d i / d t+5 i=10, i(0)=4\), \(d i(0) / d t=-2\)