1. Find the Laplace transform for $\displaystyle f\left(t\right)=e^{t}\,\sinh \left(3\,t\right)$

    $\displaystyle {{3}\over{s^2-2\,s-8}}$ $\displaystyle {{s}\over{s^2-2\,s-8}}-{{1}\over{s^2-2\,s-8}}$ $\displaystyle {{6\,s}\over{s^4-18\,s^2+81}}$ $\displaystyle {{2\,s^2}\over{s^4-18\,s^2+81}}-{{1}\over{s^2-9}}$

  2. Find the inverse transform for $\displaystyle F\left(s\right)={{1}\over{\left(s+4\right)^4}}$

    $\displaystyle {{t^3\,e^{4\,t}}\over{6}}$ $\displaystyle 2\,t^2\,e^{4\,t}+t\,e^{4\,t}$ $\displaystyle {{t^3\,e^ {- 4\,t }}\over{6}}$ $\displaystyle t\,e^ {- 4\,t }-2\,t^2\,e^ {- 4\,t }$

  3. Find the Laplace transform for $\displaystyle f\left(t\right)=t^3\,e^ {- 2\,t }$

    $\displaystyle {{2}\over{\left(s+2\right)^3}}$ $\displaystyle {{1}\over{s+2}}$ $\displaystyle {{6}\over{\left(s-2\right)^4}}$ $\displaystyle {{2}\over{\left(s-2\right)^3}}$ $\displaystyle {{6}\over{\left(s+2\right)^4}}$

  4. Find the inverse transform for $\displaystyle F\left(s\right)={{s}\over{s^2+1}}$

    $\displaystyle \cos t$ $\displaystyle \sin t$ $\displaystyle {{\sin t}\over{2}}-{{t\,\cos t}\over{2}}$ $\displaystyle {{t\,\sin t}\over{2}}$

  5. Find the Laplace transform for $\displaystyle f\left(t\right)=e^{t}\,\cosh \left(2\,t\right)$

    $\displaystyle {{2}\over{s^2-2\,s+5}}$ $\displaystyle {{s-1}\over{s^2-2\,s-3}}$ $\displaystyle {{3}\over{s^2-2\,s+10}}$ $\displaystyle {{s-1}\over{s^2-2\,s-8}}$


Department of Mathematics
Last modified: 2025-07-05