Question and Answers Forum

All Questions      Topic List

Differential Equation Questions

Previous in All Question      Next in All Question      

Previous in Differential Equation      Next in Differential Equation      

Question Number 131315 by Engr_Jidda last updated on 03/Feb/21

Given that y_n (t)=ϱ^t B_n sin((nπ)/4)t Evaluate ∫_0 ^4 [y_n (t)]^2 dt

$${Given}\:{that}\:{y}_{{n}} \left({t}\right)=\varrho^{{t}} {B}_{{n}} {sin}\frac{{n}\pi}{\mathrm{4}}{t} \\ $$$${Evaluate}\:\int_{\mathrm{0}} ^{\mathrm{4}} \left[{y}_{{n}} \left({t}\right)\right]^{\mathrm{2}} {dt} \\ $$

Answered by Ar Brandon last updated on 03/Feb/21

I=∫_0 ^4 ϱ^(2t) B_n sin^2 ((nπ)/4)t=((4B_n )/(nπ))∫_0 ^(nπ) e^((8u)/(nπ)) sin^2 udu =((2B_n )/(nπ))∫_0 ^(nπ) e^((8u)/(nπ)) (1−cos2u)du

$$\mathcal{I}=\int_{\mathrm{0}} ^{\mathrm{4}} \varrho^{\mathrm{2t}} {B}_{\mathrm{n}} \mathrm{sin}^{\mathrm{2}} \frac{\mathrm{n}\pi}{\mathrm{4}}\mathrm{t}=\frac{\mathrm{4}{B}_{\mathrm{n}} }{\mathrm{n}\pi}\int_{\mathrm{0}} ^{\mathrm{n}\pi} \mathrm{e}^{\frac{\mathrm{8u}}{\mathrm{n}\pi}} \mathrm{sin}^{\mathrm{2}} \mathrm{udu} \\ $$$$\:\:\:=\frac{\mathrm{2}{B}_{\mathrm{n}} }{\mathrm{n}\pi}\int_{\mathrm{0}} ^{\mathrm{n}\pi} \mathrm{e}^{\frac{\mathrm{8u}}{\mathrm{n}\pi}} \left(\mathrm{1}−\mathrm{cos2u}\right)\mathrm{du} \\ $$

Commented by Ar Brandon last updated on 03/Feb/21

Applying integration by-parts we′ll get result.

$$\mathrm{Applying}\:\mathrm{integration}\:\mathrm{by}-\mathrm{parts}\: \\ $$$$\mathrm{we}'\mathrm{ll}\:\mathrm{get}\:\mathrm{result}. \\ $$

Commented by Engr_Jidda last updated on 03/Feb/21

thank you sir

$${thank}\:{you}\:{sir} \\ $$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com