Full solution: Using the Laplace transform and showing the details, find the current

Using the Laplace transform and showing the details, find the current i(t) in the circuit in Fig. 126 with \(R=10 \Omega\) and \(C=10^{-2} \mathrm{~F}\) where the current at t = 0 is assumed to be zero, and: v = 0 if t < 4 and \(14 \cdot 10^{6} e^{-3 t} \mathrm{~V}\) if t > 4 Text Transcription: R=10 Omega C=10^-2 F 14 times 10^6 e^-3t V

Using the Laplace transform and showing the details, find the current i(t) in the circuit in Fig. 127, assuming i(0) = 0 and: \(R=10 \Omega, L=0.5 \mathrm{H}, v=200 t \mathrm{~V}\) if 0 < t < 2 and 0 if t > 2 Text Transcription: R=10 Omega, L=0.5 H, v=200 t V

Using the Laplace transform and showing the details, find the current i(t) in the circuit in Fig. 127, assuming i(0) = 0 and: \(R=1 \mathrm{k} \Omega(=1000 \Omega), L=1 \mathrm{H}, v=0\) if \(0<t<\pi\), \(40 \sin t \mathrm{~V} \text { if } t>\pi\) Text Transcription: R=1 k Omega (=1000 Omega), L=1 H, v=0 0 < t < pi 40 sin t V if t > pi

Using the Laplace transform and showing the details, find the current i(t) in the circuit in Fig. 127, assuming i(0) = 0 and: \(R=25 \Omega, L=0.1 \mathrm{H}, v=490 e^{-5 t} \mathrm{~V}\) if 0 < t < 1 and 0 if t > 1 Text Transcription: R=25 Omega, L=0.1 H, v=490 e^-5t V

Usmg the Laplace transform and showing the details, find the current i(t) in the circuit in Fig. 128, assuming zero initial current and charge on the capacitor and: \(L=1 \mathrm{H}, C=0.25 \mathrm{~F}, v=200\left(t-\frac{1}{3} t^{3}\right) \mathrm{V}\) if 0 < t < 1 and 0 if t > 1 Text Transcription: L=1 H, C=0.25 F, v=200(t-1/3t^3)V

Usmg the Laplace transform and showing the details, find the current i(t) in the circuit in Fig. 128, assuming zero initial current and charge on the capacitor and: \(L=1 \mathrm{H}, C=10^{-2} \mathrm{~F}, v=-9900 \cos t \mathrm{~V}\) if \(\pi<t<3 \pi\) and 0 otherwise Text Transcription: L=1 H, C=10^-2 F, v=-9900 cos t V pi < t < 3pi

Usmg the Laplace transform and showing the details, find the current i(t) in the circuit in Fig. 128, assuming zero initial current and charge on the capacitor and: \(L=0.5 \mathrm{H}, C=0.05 \mathrm{~F}, v=78 \sin t \mathrm{~V}\) if \(0<t<\pi\) and 0 if \(t>\pi\) Text Transcription: L=0.5 H, C=0.05 F, v=78 sin t V 0 < t < pi t > pi

Using the Laplace transform and showing the details, find the current i(t) in the circuit in Fig. 129. assuming zero initial current and charge and: \(R=2 \Omega, L=1 \mathrm{H}, C=0.5 \mathrm{~F}, v(t)=1 \mathrm{kV}\) if 0 < t < 2 and 0 if t > 2 Text Transcription: R=2 Omega. L=1 H, C=0.5 F. v(t)=1 kV

Using the Laplace transform and showing the details, find the current i(t) in the circuit in Fig. 129. assuming zero initial current and charge and: \(R=4 \Omega, L=1 \mathrm{H}, C=0.05 \mathrm{~F}, v=34 e^{-t} \mathrm{~V}\) 0 < t < 4 and 0 if t > 4 Text Transcription: R=4 Omega. L=1 H, C=0.05 F. v=34e^-t V

Using the Laplace transform and showing the details, find the current i(t) in the circuit in Fig. 129. assuming zero initial current and charge and: \(R=2 \Omega, L=1 \mathrm{H}, C=0.1 \mathrm{~F}, v=255 \sin t \mathrm{~V}\) if \(0<t<2 \pi\) and 0 if \(t>2 \pi\) Text Transcription: R=2 Omega. L=1 H, C=0.1 F. v=255 sin t V 0 < t < 2pi t > 2pi

WRITING PROJECT. Shifting Theorem. Explain and compare the different roles of the two shifting theorems, using your own formulations and examples.

Sketch or graph the given function (which is assumed to be zero outside the given interval). Represent it using unit step functions. Find its transform. Show the details of your work. \(t(0<t<1)\) Text Transcription: t(0<t<1)

Sketch or graph the given function (which is assumed to be zero outside the given interval). Represent it using unit step functions. Find its transform. Show the details of your work. \(e^{t}(0<t<2)\) Text Transcription: e^t(0<t<2)

Sketch or graph the given function (which is assumed to be zero outside the given interval). Represent it using unit step functions. Find its transform. Show the details of your work. \(\sin 3 t(0<t<\pi)\) Text Transcription: sin 3t(0<t<pi)

Sketch or graph the given function (which is assumed to be zero outside the given interval). Represent it using unit step functions. Find its transform. Show the details of your work. \(t^{2}(1<t<2)\) Text Transcription: t^2(1<t<2)

Sketch or graph the given function (which is assumed to be zero outside the given interval). Represent it using unit step functions. Find its transform. Show the details of your work. \(t^{2}(t>3)\) Text Transcription: t^2(t>3)

Sketch or graph the given function (which is assumed to be zero outside the given interval). Represent it using unit step functions. Find its transform. Show the details of your work. \(\cos \pi t(1<t<4)\) Text Transcription: cos pi t(1<t<4)

Sketch or graph the given function (which is assumed to be zero outside the given interval). Represent it using unit step functions. Find its transform. Show the details of your work. \(1-e^{-t}(0<t<\pi)\) Text Transcription: 1-e^-t(0<t<pi)

Sketch or graph the given function (which is assumed to be zero outside the given interval). Represent it using unit step functions. Find its transform. Show the details of your work. \(t(5<t<10)\) Text Transcription: t(5<t<10)

Sketch or graph the given function (which is assumed to be zero outside the given interval). Represent it using unit step functions. Find its transform. Show the details of your work. \(\sin \omega t(t>6 \pi / \omega)\) Text Transcription: sin omega t(t>pi/omega)

Sketch or graph the given function (which is assumed to be zero outside the given interval). Represent it using unit step functions. Find its transform. Show the details of your work. \(20 \cos \pi t(3<t<6)\) Text Transcription: 20 cos pi t(3 < t < 6)

Sketch or graph the given function (which is assumed to be zero outside the given interval). Represent it using unit step functions. Find its transform. Show the details of your work. \(\sinh t(0<t<2)\) Text Transcription: sinh t(0 < t < 2)

Sketch or graph the given function (which is assumed to be zero outside the given interval). Represent it using unit step functions. Find its transform. Show the details of your work. \(e^{\pi t}(2<t<4)\) Text Transcription: e^pi t (2 < t < 4

Find and sketch or graph f(t) if \(\mathscr{L}(f)\) equals: \(s e^{-s} /\left(s^{2}+\omega^{2}\right)\) Text Transcription: L(f) se^-s/(s^2+omega^2)

Find and sketch or graph f(t) if \(\mathscr{L}(f)\) equals: \(e^{-4 s} / s^{2}\) Text Transcription: L(f) e^4s/s^2

Find and sketch or graph f(t) if \(\mathscr{L}(f)\) equals: \(s^{-2}-\left(s^{-2}+s^{-1}\right) e^{-s}\) Text Transcription: L(f) s-^2-(s^-2+s^-1)e^-s

Find and sketch or graph f(t) if \(\mathscr{L}(f)\) equals: \(\left(e^{-2 \pi s}-e^{-8 \pi s}\right) /\left(s^{2}+1\right)\) Text Transcription: L(f) (e^-2 pi s - e^-8 pi s)/(s^2+1)

Find and sketch or graph f(t) if \(\mathscr{L}(f)\) equals: \(e^{-\pi s} /\left(s^{2}+2 s+2\right)\) Text Transcription: L(f) e^-pi s/(s^2+2s+2)

Find and sketch or graph f(t) if \(\mathscr{L}(f)\) equals: \(e^{-2 s} / s^{5}\) Text Transcription: L(f) e^-2s/s^5

Find and sketch or graph f(t) if \(\mathscr{L}(f)\) equals: \(\left(1-e^{-s+k}\right) /(s-k)\) Text Transcription: L(f) (1-e^-s+k)/(s-k)

Find and sketch or graph f(t) if \(\mathscr{L}(f)\) equals: \(s e^{-3 s} /\left(s^{2}-4\right)\) Text Transcription: L(f) se^-3s/(s^2-4)

Find and sketch or graph f(t) if \(\mathscr{L}(f)\) equals: \(2.5\left(e^{-3.8 s}-e^{-2.6 s}\right) / s\) Text Transcription: L(f) 2.5(e^-3.8s - e^-2.6s)/s

Using the Laplace transform and showing the details, solve: \(y^{\prime \prime}+2 y^{\prime}+2 v=0, \quad y(0)=0, y^{\prime}(0)=1\) Text Transcription: y’’+2y’+2v=0, y(0)=0, y’(0)=1

Using the Laplace transform and showing the details, solve: \(9 y^{\prime \prime}-6 y^{\prime}+y=0, \quad y(0)=3, y^{\prime}(0)=1\) Text Transcription: 9y’’-6y’+y=0, y(0)=3, y’(0)=1

Using the Laplace transform and showing the details, solve: \(y^{\prime \prime}+4 y^{\prime}+13 y=145 \cos 2 t, \quad y(0)=10, y^{\prime}(0)=14\) Text Transcription: y’’+4y’+13y=145 cos 2t, y(0)=10, y’(0)=14

Using the Laplace transform and showing the details, solve: \(y^{\prime \prime}+10 y^{\prime}+24 y=144 t^{2}, \quad y(0)=\frac{19}{12}, y^{\prime}(0)=-5\) Text Transcription: y’’+10y’+24y=144t^2, y(0)=19/22, y’(0)=-5

Using the Laplace transform and showing the details, solve: \(y^{\prime \prime}+9 y=r(t), r(t)=8 \sin t \text { if } 0<t<\pi\) and 0 if \(t>\pi ; \quad y(0)=0, y^{\prime}(0)=4\) Text Transcription: y’’+9y=r(t), r(t)=8 sin t if 0 < t < pi t>pi; y(0)=0, y’(0)=4

Using the Laplace transform and showing the details, solve: \(y^{\prime \prime}+3 y^{\prime}+2 y=r(t), r(t)=1\) if 0 < t < 1 and 0 if t > 1; y(0) = 0, y’(0) = 0 Text Transcription: y’’+3y’+2y=r(t), r(t)=1

Using the Laplace transform and showing the details, solve: y’’ + y = r(t), r(t) = t if 0 < t < 1 and 0 if t > 1; y(0) = y’(0) = 0

Using the Laplace transform and showing the details, solve: \(y^{\prime \prime}-16 y=r(t), \quad r(t)=48 e^{2 t}\) if 0 < t < 4 and \(0 \text { if } t>4; \quad y(0)=3 , \quad y^{\prime}(0)=-4\) Text Transcription: y’’-16y=r(t), r(t)=48e^2t 0 if t > 4; y(0)=3, y’(0)=-4

Using the Laplace transform and showing the details, solve: \(y^{\prime \prime}+y^{\prime}-2 y=r(t), r(t)=3 \sin t-\cos t\) if \(0<t<2 \pi\) and \(3 \sin 2 t-\cos 2 t \text { if } t>2 \pi ; y(0)=1, \quad y^{\prime}(0)=0\) Text Transcription: y’’+y’-2y=r(t), r(t)=3 sin t - cos t 0<t<2 pi 3 sin 2t - cos 2t if t>2 pi; y(0)=1, y’(0)=0

Using the Laplace transform and showing the details, solve: \(y^{\prime \prime}+8 y^{\prime}+15 y=r(t), r(t)=35 e^{2 t}\) if 0 < t < 2 and 0 if \(t>2 ; \quad y(0)=3, y^{\prime}(0)=-8\) Text Transcription: y’’+8y’+15v=r(t), r(t)=35e^2t t>2; y(0)=3, y’(0)=-8

Using the Laplace transform and showing the details, solve: (Shifted data)\(y^{\prime \prime}+4 y=8 t^{2}\) if 0 < t < 5 and 0 if t > 5; y(1) = 2 + cos 2, y’(1) = 4 - 2 sin 2 Text Transcription: y’’+4y=8t^2

Using the Laplace transform and showing the details, solve: y’’ + 2y’ + 5y = 10 sin t if \(0<t<2 \pi\) and 0 if \(t>2 \pi ; y(\pi)=1, y^{\prime}(\pi)=2 e^{-\pi}-2\) Text Transcription: 0 < t < 2pi t>2pi; y(pi)=1, y’(pi)=2e^-pi - 2

(Discharge) Using the Laplace transform, find the charge q(t) on the capacitor of capacitance C in Fig. 125 if the capacitor is charged so that its potential is \(V_{0}\) and the switch is closed at t = 0. Text Transcription: V_0

Using the Laplace transform and showing the details, find the current i(t) in the circuit in Fig. 126 with \(R=10 \Omega\) and \(C=10^{-2} \mathrm{~F}\) where the current at t = 0 is assumed to be zero, and: v(t) = 100 V if 0.5 < t < 0.6 and 0 otherwise. Why does i(t) have jumps? Text Transcription: R=10 Omega C=10^-2 F

Using the Laplace transform and showing the details, find the current i(t) in the circuit in Fig. 126 with \(R=10 \Omega\) and \(C=10^{-2} \mathrm{~F}\) where the current at t = 0 is assumed to be zero, and: v = 0 if t < 2 and 100 (t - 2) V if t > 2 Text Transcription: R=10 Omega C=10^-2 F