ODE6.3

From Example Problems
Jump to: navigation, search

Find the Laplace transform of f(t)={\begin{cases}1&0<t\leq 1\\-1&1<t\leq 2\end{cases}}\,

Use this property:

L\{g(t)\}={\frac  {\int _{0}^{P}e^{{-st}}g(t)\,dt}{1-e^{{-Ps}}}}\,, with period P.

L\{f(t)\}={\frac  {\int _{0}^{2}e^{{-st}}f(t)\,dt}{1-e^{{-2s}}}}\,

={\frac  {\int _{0}^{1}e^{{-st}}(1)\,dt+\int _{1}^{2}e^{{-st}}(-1)\,dt}{1-e^{{-2s}}}}\,

={\frac  {{\frac  {-e^{{-st}}}{s}}{\bigg |}_{{t=0}}^{1}+{\frac  {e^{{-st}}}{s}}{\bigg |}_{{t=1}}^{2}}{1-e^{{-2s}}}}\,

={\frac  {{\frac  {1}{s}}(1-2e^{{-s}}+e^{{-2s}})}{1-e^{{-2s}}}}\,

={\frac  {{\frac  {1}{s}}(1-e^{{-s}})^{2}}{(1+e^{{-s}})(1-e^{{-s}})}}\,

={\frac  {1}{s}}\left({\frac  {1-e^{{-s}}}{1+e^{{-s}}}}\right)\,

={\frac  {1}{s}}\left({\frac  {e^{{s/2}}-e^{{-s/2}}}{e^{{s/2}}+e^{{-s/2}}}}\right)\,

={\frac  {1}{s}}\tanh {\frac  {s}{2}}\,

Ordinary Differential Equations

Main Page