91Ó°ÊÓ

Find the unit impulse response to the given system. Assume \(y(0)=y^{\prime}(0)=0\). $$ y^{\prime \prime}-9 y=\delta(t) $$

Use the Laplace transform to solve the first-order initial value problems in Exercises 1-10. $$ y^{\prime}+6 y=2 t+3, y(0)=1 $$

Find the unit impulse response to the given system. Assume \(y(0)=y^{\prime}(0)=0\). $$ y^{\prime \prime}+4 y^{\prime}+5 y=\delta(t) $$

Use the Laplace transform to solve the first-order initial value problems in Exercises 1-10. $$ y^{\prime}+8 y=t^{2}, y(0)=-1 $$

Use the Laplace transform to solve the first-order initial value problems in Exercises 1-10. $$ y^{\prime}+8 y=e^{-2 t} \sin t, y(0)=0 $$

Find the unit impulse response to the given system. Assume \(y(0)=y^{\prime}(0)=0\). $$ y^{\prime \prime}+2 y^{\prime}+2 y=\delta(t) $$

Use the Laplace transform to solve the first-order initial value problems in Exercises 1-10. $$ y^{\prime}-2 y=e^{-t} \cos t, y(0)=-2 $$

Use the Laplace transform to solve the first-order initial value problems in Exercises 1-10. $$ y^{\prime}+y=t e^{t}, y(0)=-2 $$

In this exercise you will examine the effect of shifts in the time domain on the Laplace transform. (a) Sketch the graph of \(f(t)=\sin t\) in the time domain. Find the Laplace transform \(F(s)=\mathcal{L}\\{f(t)\\}(s)\). Sketch the graph of \(F\) in the \(s\)-domain on the interval \([0,2]\). (b) Sketch the graph of \(g(t)=H(t-1) \sin (t-1)\) in the time domain. Find the Laplace transform \(G(s)=\) \(\mathcal{L}\\{g(t)\\}(s)\). Sketch the graph of \(G\) in the \(s\)-domain on the interval \([0,2]\) on the same axes used to sketch the graph of \(F\). (c) Repeat the directions in part (b) for \(g(t)=H(t-\) 2) \(\sin (t-2)\). Explain why engineers like to say that "a shift in the time domain leads to an attenuation (scaling) in the \(s\)-domain."

Use the Laplace transform to solve the first-order initial value problems in Exercises 1-10. $$ y^{\prime}-4 y=e^{-2 t} t^{2}, y(0)=1 $$