Question and Answers Forum

All Questions      Topic List

Others Questions

Previous in All Question      Next in All Question      

Previous in Others      Next in Others      

Question Number 36853 by tawa tawa last updated on 06/Jun/18

$$\mathrm{Find}\mathrm{the}\mathrm{laplace}\mathrm{of}\mathrm{L}\left\{\frac{{\mathrm{e}}^{-\mathrm{at}}-{\mathrm{e}}^{-\mathrm{bt}}}{\mathrm{t}}\right\}$$

Commented by prof Abdo imad last updated on 06/Jun/18

$$L\left\{\frac{e^{-ax}-e^{-bx}}{x}\right\}={\int }_0^{\mathrm{\infty }}f\left(t\right)e^{-xt}dt$$$${\int }_0^{\mathrm{\infty }}\frac{e^{-at}-e^{-bt}}{t}{e}^{-xt}dt=\phi \left(x\right)$$$$soifa>0,b>0wehave$$$${\phi }^{{}^{\prime }}\left(x\right)={\int }_0^{\mathrm{\infty }}\frac{\partial }{\partial x}\left\{\frac{e^{-at}-e^{-bt}}{t}{e}^{-xt}\right\}dt$$$$=-{\int }_0^{\mathrm{\infty }}(e^{-at}-e^{-bt})e^{-xt}dt$$$$={\int }_0^{\mathrm{\infty }}(e^{-(b+x)t}-e^{-(a+x)t})e^{-xt}dt$$$$={\int }_0^{\mathrm{\infty }}{e}^{-(b+x)t}dt-{\int }_0^{\mathrm{\infty }}{e}^{-(a+x)t}dt$$$$=\frac{-1}{b+x}{\left[e^{-(b+x)t}\right]}_{t=0}^{\mathrm{\infty }}+\frac{1}{a+x}{\left[e^{-(a+x)t}\right]}_{t=0}^{\mathrm{\infty }}$$$$=\frac{1}{b+x}-\frac{1}{a+x}\Rightarrow \phi \left(x\right)=ln\mid b+x\mid -ln\mid a+x\mid +c$$$$=ln\mid \frac{b+x}{a+x}\mid +cbutc={lim}_{x\to +\mathrm{\infty }}\{\phi \left(x\right)-ln\left(\frac{b+x}{a+x}\right)\}$$$$\mathrm{\exists }m>0/\mid \phi \left(x\right)\mid <m{\int }_0^{\mathrm{\infty }}{e}^{-xt}dt=\frac{m}{x}\to 0(x\to +\mathrm{\infty })$$$$\Rightarrow c=0and\phi \left(x\right)=ln\left(\frac{b+x}{a+x}\right).$$

Commented by tawa tawa last updated on 06/Jun/18

$$\mathrm{God}\mathrm{bless}\mathrm{you}\mathrm{sir}$$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com