Chapter 8 Laplace Transforms

(a) \(\dst{1\over s^2}\) (b) \(\dst{1\over(s+1)^2}\) (c) \(\dst{b\over s^2-b^2}\) (d) \(\dst{-2s+5\over(s-1)(s-2)}\) (e) \(\dst{2\over s^3}\)

(a) \(\dst{{s^2+2\over\left[(s-1)^2+1\right] \left[(s+1)^2 +1\right]}}\) (b) \(\dst{2\over s(s^2+4)}\) (c) \(\dst{s^2+8\over s(s^2+16)}\) (d) \(\dst{s^2-2\over s(s^2-4)}\)

   (e) \(\dst{4s\over(s^2-4)^2}\) (f) \(\dst{1\over s^2+4}\) (g) \(\dst{{1\over\sqrt{2}}\,{s+1\over s^2+1}}\) (h) \(\dst{5s\over(s^2+4)(s^2+9)}\) (i) \(\dst{{s^3+2s^2+4s+32\over (s^2+4)(s^2+16)}}\)

(a) \(\dst{f(3-)=-1,\,f(3)=f(3+)=1}\) (b) \(\dst{f(1-)=3,\,f(1)=4,\,f(1+)=1}\)

   (c) \(\dst{f\left({\pi\over2}-\right)=1,\, f\left({\pi\over2}\right)=f\left({\pi\over2}+\right)=2,\, f(\pi-)=0,\,f(\pi)=f(\pi+)=-1}\)

   (d) \(\dst{f(1-)=1,\,f(1)=2,\,f(1+)=1,\,f(2-)=0,\,f(2)=3,\,f(2+)=6}\)

(a) \(\dst{{1 -e^{-(s+1)}\over s+1}+ {e^{-(s+2)}\over s+2}}\) (b) \(\dst{{1\over s}+e^{-4s}\left({1\over s^2}+{3\over s}\right)}\) (c) \(\dst{1-e^{-s}\over s^2}\) (d) \(\dst{1-e^{-(s-1)}\over(s-1)^2}\)

\(\dst{{\cal L}(e^{\lambda t}\cos\omega t)={(s-\lambda)^2 - \omega^2\over\left((s-\lambda)^2+\omega^2\right)^2}\; {\cal L}(e^{\lambda t}\sin\omega t)= {2\omega(s-\lambda)\over\left((s-\lambda)^2+\omega^2\right)^2}}\)

(a) \(\dst{\tan^{-1} {\omega\over s},\quad s>0}\) (b) \(\dst{{1\over 2}\ln{s^2\over s^2+\omega^2},\quad s>0}\) (c) \(\dst{{\ln{s-b\over s-a},\quad s>\max (a,b)}}\)

   (d) \(\dst{{1\over 2}\ln{s^2\over s^2-1},\quad s>1}\) (e) \(\dst{{1\over 4}\ln{s^2\over s^2-4},\quad s>2}\)

(a) \(\dst{1\over s^2}\tanh{s\over 2}\)  (b) \(\dst{1\over s}\tanh{s\over 4}\)  (c) \(\dst{{1\over s^2+1}\coth {\pi s\over 2}}\)  (d) \(\dst{1\over(s^2+1)(1-e^{-\pi s})}\)

(a) \(\dst{\frac{t^{3}e^{7t}}{2}}\)  (b) \(2e^{2t}\cos 3t\)  (c) \(\dst{e^{-2t}\over4}\sin 4t\)  (d) \(\dst{2\over3}\sin 3t\)  (e) \(t\cos t\) 

   (f) \(\dst{e^{2t}\over2}\sinh 2t\)  (g) \(\dst{2te^{2t}\over3} \sin 9t\)  (h) \(\dst{2e^{3t}\over3}\sinh 3t\)  (i) \(e^{2t} t\cos t\)

(a) \(\dst{t^2e^{7t}+{17\over6}t^3e^{7t}}\)  (b) \(\dst{e^{2t}\left(\frac{1}{6}t^3+\frac{1}{6}t^4+\frac{1}{40}t^5 \right)}\)  (c) \(\dst{e^{-3t}\left(\cos 3t+{2\over3} \sin 3t\right)}\) 

   (d) \(\dst{2\cos 3t+{1\over3}\sin 3t}\)  (e) \((1-t)e{^{-t}}\)  (f) \(\dst{\cosh 3t+{1\over3}\sinh 3t}\)  (g) \(\dst{\left(1-t-t^2-{1\over6} t^3\right)e^{-t}}\) 

   (h) \(\dst{e^t\left(2\cos 2t+{5\over2}\sin 2t\right)}\)  (i) \(1-\cos t\)  (j) \(3\cosh t+4\sinh t\)  (k) \(\dst{3e^t+4\cos 3t+{1\over3}\sin 3t}\) 

   (l) \(3te^{-2t}-2\cos 2t-3\sin 2t\)

(a) \(\dst{{1\over4}e^{2t}- {1\over4}e^{-2t}-e^{-t}}\)  (b) \(\dst{{1\over5}e^{-4t}-{41\over5}e^t+5e^{3t}}\)  (c) \(\dst{-{1\over2}e^{2t}-{13\over10}e^{-2t}\; -{1\over5}e^{3t}}\) 

   (d) \(\dst{-{2\over5}e^{-4t}-{3\over5}e^t}\)  (e) \(\dst{{3\over20}e^{2t}-{37\over12}e^{-2t}\; +{1\over3}e^t+{8\over5}e^{-3t}}\) (f) \(\dst{{39\over10}e^t+{3\over14}e^{3t}+{23\over105}e^{-4t} -{7\over3} e^{2t}}\) 

  (a) \(\dst{{4\over5}e^{-2t}-{1\over2}e^{-t}-{3\over10}\cos t} +{11\over10}\sin t\)  (b) \(\dst{{2\over5}\sin t+{6\over5}\cos t +{7\over5}e^{-t}\sin t-{6\over5}e^{-t}\cos t}\)

   (c) \(\dst{{8\over13}e^{2t}-{8\over13}e^{-t}\cos 2t+{15\over26} e^{-t}\sin 2t}\)  (d) \(\dst{{1\over2}te^t+{3\over8}e^t+e^{-2t} -{11\over8}e^{-3t}}\)

   (e) \(\dst{{2\over3}te^t+{1\over9}e^t +te^{-2t}-{1\over9}e^{-2t}}\) (f) \(\dst{-e^t+{5\over2}te^t+\cos t-{3\over2}\sin t}\)

  (a) \(\dst{{3\over5}\cos 2t+{1\over5}\sin 2t -{3\over5}\cos 3t-{2\over 15}\sin 3t}\)  (b) \(\dst{-{4\over 15}\cos t+{1\over 15}\sin t +{4\over 15}\cos 4t-{1\over60}\sin 4t}\) 

   (c) \(\dst{{5\over3}\cos t+\sin t-{5\over3} \cos 2t-{1\over2}\sin 2t}\)  (d) \(\dst{-{1\over3}\cos{t\over2}+{2\over3} \sin{t\over2}+{1\over3}\cos t-{1\over3}\sin t}\) 

   (e) \(\dst{{1\over 15}\cos{t\over4}-{8\over 15}\sin{t\over4} -{1\over 15}\cos 4t+{1\over30}\sin4t}\)  (f) \(\dst{{2\over5}\cos{t\over3}-{3\over5}\sin{t\over3} -{2\over5}\cos{t\over2}+{2\over5}\sin{t\over2}}\)

  (a) \(\dst{e^t(\cos 2t+\sin 2t)-e^{-t}\left(\cos 3t+{4\over3}\sin 3t\right)}\)  (b) \(\dst{e^{3t}\left(-\cos 2t+{3\over2}\sin 2t\right)+e^{-t}\left(\cos 2t+{1\over2}\sin 2t\right)}\)

   (c) \(\dst{e^{-2t}\left({1\over8}\cos t+{1\over 4}\sin t\right)-e^{2t}\left({1\over8}\cos 3t-{1\over 12}\sin 3t\right)}\)  (d) \(\dst{e^{2t}\left(\cos t+{1\over2}\sin t\right)-e^{3t}\left(\cos 2t-{1\over4}\sin 2t\right)}\)

   (e) \(\dst{e^t\left({1\over5}\cos t+{2\over5} \sin t\right) -e^{-t}\left({1\over5}\cos 2t+{2\over5}\sin 2t\right)}\)  (f) \(\dst{e^{t/2}\left(-\cos t+{9\over8}\sin t\right)+e^{-t/2}\left(\cos t-{1\over8}\sin t\right)}\)

(a) \(1-\cos t\)  (b) \(\dst{{e^t\over 16}(1-\cos 4t)}\)  (c) \(\dst{{4\over9}e^{2t}+{5\over9}e^{-t} \sin 3t-{4\over9}e^{-t}\cos 3t}\) (d) \(\dst{3e^{t/2}-{7\over2}e^t\sin 2t-3e^t\cos 2t}\)

   (e) \(\dst{{1\over4}e^{3t}-{1\over4}e^{-t}\cos 2t}\)  (f) \(\dst{{1\over9}e^{2t}-{1\over9}e^{-t}\cos 3t+{5\over 9}e^{-t}\sin 3t}\)

  (a) \(\dst{-{3\over 10}\sin t+{2\over5}\cos t- {3\over4}e^t+{7\over20}e^{3t}}\)  (b) \(\dst{-{3\over5}e^{-t}\sin t+{1\over5} e^{-t}\cos t-{1\over2}e^{-t}+{3\over 10}e^{t}}\) 

   (c) \(\dst{-{1\over 10}e^t\sin t-{7\over 10}e^t \cos t+{1\over5} e^{-t}+{1\over2}e^{2t}}\)  (d) \(\dst{-{1\over2}e^t+{7\over 10}e^{-t}-{1\over 5}\cos 2t+{3\over5}\sin 2t}\)

   (e) \(\dst{{3\over 10}+{1\over 10}e^{2t}+{1\over 10}e^t\sin 2t-{2\over5}e^t\cos 2t}\)  (f) \(\dst{-{4\over9}e^{2t}\cos 3t+{1\over3} e^{2t}\sin 3t-{5\over9} e^{2t}+e^t}\)

\(\dst{{1\over a}e^{{b\over a}t}f\left({t\over a}\right)}\)

\(y=\dst{{1\over6}e^t-{9\over 2}e^{-t}+{16\over3}e^{-2t}}\)

\(y=\dst{-{1\over3}+{8\over15}e^{3t}+{4\over 5}e^{-2t}}\)

\(y=\dst{-{23\over15}e^{-2t}+{1\over3}e^t +{1\over5}e^{3t}}\)

\(y=\dst{-{1\over4}e^{2t}+{17\over 20}e^{-2t}+{2\over5}e^{3t}}\)

\(y=\dst{{11\over15}e^{-2t}+{1\over 6}e^t+{1\over10}e^{3t}}\)

\(y=e^t+2e^{-2t}-2e^{-t}\)

\(y=\dst{{5\over3}\sin t-{1\over3}\sin2t}\)

\(y=4e^t-4e^{2t}+e^{3t}\)

\(y=\dst{-{7\over2}e^{2t}+{13\over 3}e^t+{1\over6}e^{4t}}\)

\(y=\dst{{5\over2}e^t-4e^{2t}+{1\over 2}e^{3t}}\)

\(y=\dst{{1\over3}e^t-2e^{-t}+{5\over3}e^{-2t}}\)

\(y=2-e^{-2t}+e^t\)

\(y=\dst{1-\cos2t+{1\over2}\sin2t}\)

\(y=\dst{-{1\over3}+{8\over15}e^{3t} +{4\over5}e^{-2t}}\)

\(y=\dst{{1\over6}e^t-{2\over 3}e^{-2t}+{1\over2}e^{-t}}\)

\(y=-1+e^t+e^{-t}\)

\(y=\cos2t-\sin2t+\sin t\)

\(y=\dst{{7\over3}-{7\over2}e^{-t} +{1\over6}e^{3t}}\)

\(y=1+\cos t\)

\(y=t+\sin t\)

\(y=t-6\sin t+\cos t+\sin2t\)

\(y=e^{-t}+4e^{-2t}-4e^{-3t}\)

\(y=-3\cos t-2\sin t+ e^{-t}(2+5t)\)

\(y=-\sin t-2\cos t+3e^{3t}+e^{-t}\)

\(y=(3t+4)\sin t-(2t+6)\cos t\)

\(y=-(2t+2)\cos2t+\sin2t+3\cos t\)

\(y=e^t(\cos t-3\sin t)+e^{3t}\)

\(y=-1+t+e^{-t} (3\cos t-5\sin t)\)

\(y=4\cos t-3\sin t-e^{t}(3\cos t-8\sin t)\)

\(y=e^{-t}-2e^t+e^{-2t} (\cos 3t-{11/3} \sin3t)\)

\(y=e^{-t}(\sin t-\cos t)+e^{-2t}(\cos t +4\sin t)\)

\(y=\dst{{1\over5}e^{2t}-{4\over3}e^t +{32\over15}e^{-t/2}}\)

\(y=\dst{{1\over7}e^{2t}-{2\over 5}e^{t/2}+{9\over35}e^{-t/3}}\)

\(y=\dst{e^{-t/2}(5\cos(t/2)-\sin(t/2))+2t-4}\)

\(y=\dst{1\over17}\left(12\cos t+20\sin t-3e^{t/2}(4\cos t+\sin t )\right)\).

\(y=\dst{{e^{-t/2}\over10}(5t+26)-{1\over5}(3\cos t+\sin t)}\)

\(y=\dst{1\over100}\left(3e^{3t}-e^{t/3}(3+310t)\right)\)

\( 1+u(t-4)(t-1)\);  \(\dst{{1\over s}+e^{-4s}\left({1\over s^2}+{3\over s}\right)}\)

\(t+u(t-1)(1-t)\);  \(\dst{1-e^{-s}\over s^2}\)

\(2t-1-u(t-2)(t-1)\);  \(\dst{\left({2\over s^2}-{1\over s}\right)-e^{-2s}\left({1\over s^2} +{1\over s}\right)}\)

\(1+u(t-1)(t+1)\);  \(\dst{{1\over s} +e^{-s}\left({1\over s^2}+{2\over s}\right)}\)

\(t-1+u(t-2)(5-t)\);  \(\dst{{1\over s^2}-{1\over s}-e^{-2s}\left({1\over s^2}-{3\over s}\right)}\)

\(t^2\left(1-u(t-1)\right)\);  \(\dst{ {2\over s^3}-e^{-s}\left({2\over s^3}+{2\over s^2}+{1\over s}\right)}\)

\(u(t-2)(t^2+3t)\);  \(\dst{ e^{-2s}\left({2\over s^3}+{7\over s^2}+{10\over s}\right)}\)

\(t^2+2+u(t-1)(t-t^2-2)\);  \(\dst{{2\over s^3} +{2\over s}-e^{-s}\left({2\over s^3}+{1\over s^2}+{2 \over s}\right)}\)

\(te^t+u(t-1)(e^t-te^t)\);  \(\dst{1-e^{-(s-1)}\over (s-1)^2}\)

\(e^{-t}+u(t-1)(e^{-2t}-e^{-t})\) ;  \(\dst{{1-e^{-(s+1)}\over s+1}+{e^{-(s+2)}\over s+2}}\)

\(-t+2u(t-2)(t-2)-u(t-3)(t-5)\);  \(\dst{-{1\over s^2}+ {2e^{-2s}\over s^2}+e^{-3s}\left({2\over s}-{1\over s^2}\right)}\)

\(\left[u(t-1)-u(t-2)\right]t\) ;  \(\dst{ e^{-s}\left({1\over s^2}+{1\over s}\right)-e^{-2s}\left({1\over s^2} +{2\over s}\right)}\)

\(t+u(t-1)(t^2-t)-u(t-2)t^2\);  \(\dst{{1\over s^2}+ e^{-s}\left({2\over s^3}+{1\over s^2}\right) -e^{-2s}\left({2\over s^3}+{4\over s^2}+{4\over s}\right)}\)

\(t+u(t-1)(2-2t)+u(t-2)(4+t)\);  \(\dst{{1\over s^2} -2{e^{-s}\over s^2}+e^{-2s}\left({1\over s^2}+{6\over s}\right)}\)

\(\dst{\sin t+ u(t-\pi/2)\sin t+u(t-\pi) (\cos t-2\sin t)}\);  \(\dst{{1+ e^{-{\pi\over 2}s}s-e^{-\pi s}(s-2)\over s^2+1}}\)

\(\dst{2-2u(t-1)t+u(t-3)(5t-2)}\);  \(\dst{{2\over s}-e^{-s}\left({2\over s^2}+{2\over s}\right)+e^{-3s}\left({5\over s^2}+{13\over s}\right)}\)

\(\dst{3+u(t-2)(3t-1)+u(t-4)(t-2)}\);  \(\dst{{3\over s}+e^{-2s}\left({3\over s^2}+{5\over s}\right)+e^{-4s}\left({1\over s^2}+{2\over s}\right)}\)

\(\dst{(t+1)^2+u(t-1)(2t+3)}\);  \(\dst{{2\over s^3 }+{2\over s^2}+{1\over s}+e^{-s}\left({2\over s^2}+{5\over s}\right)}\)

\(u(t-2)e^{2(t-2)}= \dst{\left\{\begin{array}{cl} 0,&0\le t<2, \\[6pt] e^{2(t-2)},&t\ge2.\end{array}\right.}\)

\(u(t-1)\left(1-e^{-(t-1)}\right)= \dst{\left\{\begin{array}{cl} 0,&0\le t<1,\\[6pt] 1-e^{-(t-1)},&t\ge1.\end{array}\right.}\)

\(\dst{u(t-1){(t-1)^2 \over 2}+u(t-2)(t-2)}= \dst{\left\{\begin{array}{cl} 0,&0 \le t< 1,\\[6pt]\dst{(t-1)^2\over 2},&1\le t<2,\\[6pt] \dst{t^2-3\over 2},&t\ge2.\end{array}\right.}\)

\(2+t+u(t-1)(4-t)+u(t-3)(t-2)= \dst{ \left\{\begin{array}{cl} 2+t,& 0\le t<1,\\[6pt] 6,&1\le t<3,\\[6pt] t+4,&t\ge 3.\end{array}\right.}\)

\(\dst{5-t+u(t-3)(7t-15)+ {3\over 2} u(t-6)(t-6)^2}= \dst{\left\{\begin{array}{cl} 5-t,&0\le t<3,\\[6pt] 6t-10,&3\le t<6,\\[6pt] 44-12t+{3\over2}t^2, &t\ge6. \end{array}\right.}\)

\(u(t-\pi)e^{-2(t-\pi)} (2\cos t- 5\sin t)= \dst{\left\{\begin{array}{cl} 0, &0\le t<\pi,\\[6pt] e^{-2(t-\pi)}(2\cos t-5\sin t),&t\ge\pi. \end{array}\right.}\)

\(\dst{1-\cos t+ u(t-\pi/2) (3\sin t+\cos t)}= \dst{\left\{\begin{array}{cl} 1-\cos t,&0\le t<\dst{\pi\over 2},\\[6pt] 1+3\sin t,&t\ge \dst{\pi\over 2}.\end{array}\right.}\)

\(\dst{u(t-2)\left (4e^{-(t-2)}-4e^{2(t-2)}+2e^{(t-2)}\right)}= \dst{\left\{\begin{array}{cl} 0,&0\le t<2,\\[6pt]4e^{-(t-2)}-4e^{2(t-2)}+2e^{(t-2)},&t\ge 2.\end{array}\right.}\)

\(\dst{1+t+u(t-1)(2t+1)+u(t-3)(3t-5)}= \dst{ \left\{\begin{array}{cl} t+1,&0\le t<1,\\[6pt] 3t+2,&1\le t<3,\\[6pt]6t-3,&t\ge3.\end{array}\right.}\)

\(\dst{1-t^2+u(t-2)\left(-{t^2\over2}+2t+1\right) +u(t-4)(t-4)}= \dst{\left\{\begin{array}{cl} 1-t^2,&0\le t<2\\[6pt]-\dst{3t^2\over2} +2t+2,&2\le t<4,\\[6pt]-\dst{3t^2\over2}+3t-2,&t\ge 4.\end{array}\right.}\)

\(\dst{e^{-\tau s}\over s}\)

For each \(t\) only finitely many terms are nonzero.

\(\dst{1+\sum_{m=1}^\infty u(t-m);\; {1\over s(1-e^{-s})}}\)

\(\dst{1+2\sum_{m=1}^\infty (-1)^mu(t-m);\; {1\over s};\; {1-e^{-s}\over 1+e^{-s}}}\)

\(\dst{1+\sum_{m=1}^\infty(2m+1)u(t-m);\; {e^{-s}(1+e^{-s})\over s(1-e^{-s})^2}}\)

\(\dst{\sum_{m=1}^\infty(-1)^m(2m-1)u(t-m);\; {1\over s}{(1-e^s)\over(1+e^s)^2}}\)

\(y=3(1-\cos t)-3u(t-\pi)(1+\cos t)\)

\(y=3-2\cos t+2u(t-4)\left(t-4-\sin (t-4)\right)\)

\(y=\dst{-{15\over2}+{3\over 2}e^{2t}-2t+{u(t-1)\over2} (e^{2(t-1)}-2t+1)}\)

\(y=\dst{{1\over2}e^t+{13\over6}e^{-t}+ {1\over 3}e^{2t} +u(t-2)\left(-1+{1\over 2}e^{t-2}+{1\over2}e^{-(t-2)}+ {1\over2}e^{t+2}-{1\over6} e^{-(t-6)}-{1\over3}e^{2t}\right)}\)

\(y=\dst{-7e^t+4e^{2t}+u(t-1)\left({1\over 2}-e^{t-1}+{1\over2}e^{2(t-1)}\right) -2u(t-2)\left({1\over2}- e^{t-2}+{1\over2}e^{2(t-2)}\right)}\)

\(y=\dst{{1\over3}\sin2t-3\cos2t+{1\over3}\sin t -2 u(t-\pi)\left({1\over3}\sin t+{1\over6}\sin2t\right) +u(t-2\pi)\left({1\over3}\sin t-{1\over6}\sin2t\right)}\)

\(y=\dst{{1\over4}-{31\over12} e^{4t}+{16\over3} e^t +u(t-1)\left({2\over3}e^{t-1}-{1\over6}e^{4(t-1)}-{1\over2} \right) +u(t-2)\left({1\over4}+{1 \over12} e^{4(t-2)}-{1\over3} e^{t-2}\right)}\)

\(y=\dst{{1\over8}\left(\cos t-\cos 3t\right) - {1\over8} u\left( t-{3\pi\over2}\right)\left(\sin t-\cos t+ \sin 3t-{1\over3}\cos 3t\right)}\)

\(y=\dst{ {t\over4}-{1\over8}\sin2t +{1\over8}u\left(t-{\pi\over2}\right) (\pi\cos2t-\sin2t+2\pi-2t)}\)

\(y=\dst{t-\sin t-2u(t-\pi)(t+\sin t+\pi\cos t)}\)

\(y=\dst{u(t-2)\left(t-{1\over2}+{e^{2(t-2)}\over2} -2e^{t-2}\right)}\)

\(y=\dst{t+\sin t+\cos t-u(t-2\pi)(3t-3\sin t-6\pi\cos t)}\)

\(y=\dst{{1\over2}+{1\over2}e^{-2t}-e^{-t}+u(t-2)\left(2 e^{-(t-2)}-e^{-2(t-2)}-1\right)}\)

\(y=\dst{-{1\over3}-{1\over6}e^{3t} +{1\over2}e^t+u(t-1)\left({2\over3}+{1\over3}e^{3(t-1)} -e^{t-1}\right)}\)

\(y=\dst{{1\over4}\left(e^t+e^{-t}(11+6t)\right) +u(t-1)(te^{-(t-1)}-1)}\)

\(y=e^t-e^{-t}-2te^{-t}-u(t-1)\left(e^t-e^{-(t-2)}-2(t-1)e^{-(t-2)}\right)\)

\(y=\dst{te^{-t}+e^{-2t}+u(t-1)\left(e^{-t}(2-t)-e^{-(2t-1)}\right)}\)

\(y=y=\dst{t^2e^{2t}\over2}-te^{2t}-u(t-2)(t-2)^2e^{2t}\)

\(y=\dst{{t^4\over12}+1-{1\over12}u(t-1)(t^4+2t^3-10t+7)+ {1\over6}u(t-2)(2t^3+3t^2-36t+44)}\)

\(y=\dst{{1\over2}e^{-t}(3\cos t+\sin t)+{1\over2}}\)

   \(\dst{-u(t-2\pi)\left( e^{-(t-2\pi)}\left((\pi-1)\cos t+{2\pi-1\over2}\sin t\right)+ 1-{t\over2}\right)}\)

   \(\dst{-{1\over2}u(t-3\pi)\left(e^{-(t-3\pi)}(3\pi\cos t+(3\pi+1)\sin t )+t\right)}\)

\(y=\dst{{t^2\over2}+\sum_{m=1}^\infty u(t-m){(t-m)^2\over2}}\)

(a) \(y=\dst{\left\{\begin{array}{cll}2m+1-\cos t,&2m\pi\le t<(2m+1)\pi&(m=0,1,\dots)\\[3pt] 2m,&(2m-1)\pi\le t<2m\pi&(m=1,2,\dots)\end{array}\right.}\)

   (b) \(y=(m+1)(t-\sin t-m\pi\cos t),\; 2m\pi\le t<(2m+2)\pi\quad (m=0,1,\dots)\)

   (c) \(y=(-1)^m-(2m+1)\cos t,\; m\pi\le t<(m+1)\pi\quad(m=0,1,\dots)\)

   (d) \(y=\dst{{e^{m+1}-1\over2(e-1)}(e^{t-m}+e^{-t})-m-1,\quad m\le t<m+1 \; (m=0,1\dots)}\)

   (e) \(y=\dst{\left(m+1-\left({e^{2(m+1)\pi}-1\over e^{2\pi}-1}\right)e^{-t}\right)\sin t\; 2m\pi\le t<2(m+1)\pi\quad (m=0,1,\dots)}\)

   (f) \(y=\dst{{m+1\over2}-e^{t-m}{e^{m+1}-1\over e-1}+{1\over2}e^{2(t-m)}{e^{2m+2}-1\over e^2-1}, \; m\le t<m+1\quad(m=0,1,\dots)}\)

(a) \(\dst{{1\over 2}\int_0^t\tau\sin 2(t-\tau)\, d\tau}\) (b) \(\dst{\int_0^t e^{-2\tau}\cos 3(t-\tau)\,d\tau}\)

   (c) \(\dst{{1\over 2}\int_0^t\sin 2\tau \cos 3(t-\tau)\,d\tau\mbox{ or } {1\over 3}\int_0^t\sin 3\tau \cos 2 (t-\tau)\,d\tau}\) (d) \(\dst{\int_0^t\cos\tau\sin (t-\tau)\, d\tau}\)

   (e) \(\dst{\int_0^t e^{a\tau}d\tau}\) (f) \(\dst{e^{-t}\int_0^t\sin (t-\tau)\,d\tau}\) (g) \(\dst{e^{-2t}\int_0^t\tau e^\tau\sin (t-\tau)\,d\tau}\)

   (h) \(\dst{{e^{-2t}\over 2}\int_0^t\tau^2 (t-\tau)e^{3\tau}\,d\tau}\) (i) \(\dst{\int_0^t (t-\tau)e^\tau\cos\tau\,d\tau}\) (j) \(\dst{\int_0^t e^{-3\tau}\cos\tau\cos 2 (t-\tau)\,d\tau}\)

   (k) \(\dst{{1\over 4!5!}\int_0^t\tau^4 (t-\tau)^5 e^{3\tau}\,d\tau}\) (l) \(\dst{{1\over 4}\int_0^t\tau^2 e^\tau \sin 2(t-\tau)\,d\tau}\)

   (m) \(\dst{{1\over 2}\int_0^t\tau(t-\tau)^2 e^{2(t-\tau)}\,d\tau}\) (n) \(\dst{{1\over 5!6!}\int_0^t (t-\tau)^5 e^{2(t-\tau)}\tau^6\,d\tau}\)

   (a) \(\dst{as\over(s^2+a^2)(s^2+b^2)}\)  (b) \(\dst{a\over(s-1)(s^2+a^2)}\)  (c) \(\dst{as\over(s^2-a^2)^2}\)  (d) \(\dst{2\omega s(s^2-\omega^2)\over (s^2+\omega^2)^4}\)

     (e) \(\dst{(s-1)\omega\over\left((s-1)^2+ \omega^2\right)^2}\)  (f) \(\dst{2\over(s-2)^3 (s-1)^2}\)  (g) \(\dst{s+1\over(s+2)^2\left[(s+1)^2+ \omega^2\right]}\)

     (h) \(\dst{1\over(s-3)\left((s-1)^2-1\right)}\)  (i) \(\dst{2\over(s-2)^2(s^2+4)}\)  (j) \(\dst{6\over s^4(s-1)}\)  (k) \(\dst{{3\cdot 6!\over s^7\left[(s+1)^2+ 9\right]}}\)

   (l) \(\dst{12\over s^7}\)  (m) \(\dst{{2\cdot 7!\over s^8\left[(s+1)^2+ 4\right]}}\)  (n) \(\dst{48\over s^5(s^2+4)}\)

  (a) \(\dst{y={2\over\sqrt{5}}\int_0^t f(t-\tau)e^{-3\tau/2}\sinh{\sqrt{5}\tau\over 2}\,d\tau}\)  (b) \(\dst{y={1\over2}\int_0^t f(t-\tau)\sin 2\tau\,d\tau}\)

     (c) \(\dst{y=\int_0^t\tau e^{-\tau}f(t-\tau)\; d\tau}\)  (d) \(\dst{y(t)=-{1\over k}\sin kt +\cos kt+ {1\over k}\int_0^t f(t-\tau)\sin k\tau\,d\tau}\)

     (e) \(\dst{y=-2te^{-3t}+\int_0^t\tau e^{-3\tau}f(t-\tau)\,d\tau}\) (f) \(\dst{y={3\over 2}\sinh 2t+{1\over 2}\int_0^t f(t-\tau)\sinh 2\tau\,d\tau}\)

     (g) \(\dst{y=e^{3t}+\int_0^t(e^{3\tau} -e^{2\tau})f(t-\tau)\,d\tau}\)  (h) \(\dst{y={k_1\over\omega} \sin\omega t+k_0\cos\omega t+{1\over\omega}\int_0^t f(t-\tau)\sin \omega\tau\,d\tau}\)

(a) \(y=\sin t\)(b) \(y=te^{-t}\)  (c) \(y=1+2te^t\)  (d) \(\dst{y=t+{t^2\over 2}}\)

   (e) \(\dst{y=4+{5\over 2}t^2+{1\over 24}t^4}\)  (f) \(\dst{y=1-t}\)

  (a) \(\dst{7!8!\over 16!}t^{16}\)  (b) \(\dst{13!7!\over 21!} t^{21}\)  (c) \(\dst{6!7!\over 14!}t^{14}\)  (d) \(\dst{1\over2}(e^{-t}+\sin t-\cos t)\)  (e) \(\dst{1\over3}(\cos t-\cos 2t)\)

\(y=\dst{\frac{1}{2}e^{2t}-4e^{-t}+\frac{11}{2}e^{-2t}+2u(t-1)(e^{-(t-1)}-e^{-2(t-1)})}\)

\(\dst{y=2e^{-2t}+5e^{-t}+\frac{5}{3}u(t-1)(e^{(t-1)}-e^{-2(t-1)})}\)

\(\dst{y=\frac{1}{6}e^{2t}-\frac{2}{3}e^{-t}-\frac{1}{2}e^{-2t}+\frac{5}{2}u(t-1)\sinh2(t-1)}\)

\(\dst{y=\frac{1}{8}(8\cos t-5\sin t-\sin3t)-2u(t-\pi/2)\cos t}\)

\(\dst{y=1-\cos2t+\frac{1}{2}\sin2t+\frac{1}{2}u(t-3\pi)\sin2t}\)

\(\dst{y=4e^{t}+3e^{-t}-8+2u(t-2)\sinh(t-2)}\)

\(\dst{y=\frac{1}{2}e^{t}-\frac{7}{2}e^{-t}+2+3u(t-6)(1-e^{-(t-6)})}\)

\(\dst{y=e^{2t}+7\cos 2t-\sin2t-\frac{1}{2}u(t-\pi/2)\sin 2t}\)

\(\dst{y=\frac{1}{2}(1+e^{-2t})+u(t-1)(e^{-(t-1)}-e^{-2(t-1)})}\)

\(\dst{y=\frac{1}{4}e^{t}+\frac{1}{4}e^{-t}(2t-5)+2u(t-2)(t-2)e^{-(t-2)}}\)

\(\dst{y=\frac{1}{6}(2\sin t+5\sin 2t)-\frac{1}{2}u(t-\pi/2)\sin2t}\)

\(\dst{y=e^{-t}(\sin t-\cos t)-e^{-(t-\pi)}\sin t-3u(t-2\pi)e^{-(t-2\pi)}\sin t}\)

\(\dst{y=e^{-2t}\left(\cos 3t+\frac{4}{3}\sin3t\right)-\frac{1}{3} u(t-\pi/6)e^{-2(t-\pi/6)}\cos3t-\frac{2}{3}u(t-\pi/3)e^{-2(t-\pi/3)}\sin3t}\)

\(\dst{y=\frac{7}{10}e^{2t}-\frac{6}{5}e^{-t/2}-\frac{1}{2}+\frac{1}{5}u(t-2)} (e^{2(t-2)}-e^{-(t-2)/2})\)

\(\dst{y=\frac{1}{17}(12\cos t+20\sin t)+\frac{1}{34}e^{t/2}(10\cos t-11\sin t) -u(t-\pi/2)e^{(2t-\pi)/4}\cos t}\)

\(+u(t-\pi)e^{(t-\pi)/2}\sin t\)

\(\dst{y=\frac{1}{3}(\cos t-\cos2t-3\sin t)-2u(t-\pi/2)\cos t+3u(t-\pi)\sin t}\)

\(\dst{y=e^{t}-e^{-t}(1+2t)-5u(t-1)\sinh(t-1)+3u(t-2)\sinh (t-2)}\)

\(\dst{y=\frac{1}{4}(e^{t}-e^{-t}(1+6t))-u(t-1)e^{-(t-1)}+2u(t-2)e^{-(t-2)})}\)

\(\dst{y=\frac{5}{3}\sin t-\frac{1}{3}\sin2t+\frac{1}{3}u(t-\pi)(\sin2t+2\sin t) +u(t-2\pi)\sin t}\)

\(\dst{y=\frac{3}{4}\cos2t-\frac{1}{2}\sin2t+\frac{1}{4}+\frac{1}{4}u(t-\pi/2) (1+\cos2t)+\frac{1}{2}u(t-\pi)\sin2t+\frac{3}{2}u(t-3\pi/2)\sin 2t}\)

\(y=\cos t-\sin t\)  

\(\dst{y=\frac{1}{4}(8e^{3t}-12e^{-2t})}\)

\(\dst{y=5(e^{-2t}-e^{-t})}\)  

\(y=e^{-2t}(1+6t)\)

\(\dst{y=\frac{1}{4}e^{-t/2}(4-19t)}\)

\(\dst{y=(-1)^{k}m\omega_{1}Re^{-c\tau/2m}\delta(t-\tau)}\) if \(\omega_{1}\tau-\phi=(2k+1)\pi/2\) (\(k=\) integer)

(a) \(\dst{y=\frac{(e^{m+1}-1)(e^{t-m}-e^{-t})}{2(e-1)}}\),   \(m\le t<m+1\), (\(m=0\), \(1\), …)

  (b) \(y=(m+1)\sin t\),  \(2m\pi\le t<2(m+1)\pi\),  (\(m=0\), \(1\), …)

  (c) \(\dst{y=e^{2(t-m)}\frac{e^{2m+2}-1}{e^{2}-1}-e^{(t-m)}\frac{e^{m+1}-1}{e-1}}\),  \(m\le t<m+1\)  (\(m=0\), \(1\), …)

  (d) \(y= \begin{cases} \quad\, 0,&\quad 2m\pi\le t<(2m+1)\pi,\\ -\sin t,& (2m+1)\pi\le t <(2m+2)\pi, \end{cases}\)   (\(m=0\), \(1\),…)