Question Number 49358 by peter frank last updated on 06/Dec/18
Answered by tanmay.chaudhury50@gmail.com last updated on 07/Dec/18
$$\left.\mathrm{1}\right)\mathrm{9}\frac{{d}^{\mathrm{2}} {x}}{{dt}^{\mathrm{2}} }+\mathrm{6}\frac{{dx}}{{dt}}+{x}=−\mathrm{50}{sint} \\ $$$${step}\mathrm{1} \\ $$$$\mathrm{9}\frac{{d}^{\mathrm{2}} {x}}{{dt}^{\mathrm{2}} }+\mathrm{6}\frac{{dx}}{{dt}}+{x}=\mathrm{0} \\ $$$${x}={e}^{\alpha{t}} \:{is}\:{a}\:{solutiin} \\ $$$$\mathrm{9}\left(\alpha^{\mathrm{2}} {e}^{\alpha{t}} \right)+\mathrm{6}\left(\alpha{e}^{\alpha{t}} \right)+{e}^{\alpha{t}} =\mathrm{0} \\ $$$${e}^{\alpha{t}} \neq\mathrm{0} \\ $$$$\mathrm{9}\alpha^{\mathrm{2}} +\mathrm{6}\alpha+\mathrm{1}=\mathrm{0} \\ $$$$\left(\mathrm{3}\alpha+\mathrm{1}\right)^{\mathrm{2}} =\mathrm{0} \\ $$$$\alpha=\frac{−\mathrm{1}}{\mathrm{3}} \\ $$$${C}.{F}\:={Ate}^{\frac{−{t}}{\mathrm{3}}} +{Be}^{\frac{−{t}}{\mathrm{3}}} ={e}^{\frac{−{t}}{\mathrm{3}}} \left({At}+{B}\right) \\ $$$${P}.{I}\:{calculation}… \\ $$$${D}=\frac{{d}}{{dt}}\:\:\:\:\:{D}^{\mathrm{2}} =\frac{{d}^{\mathrm{2}} }{{dt}^{\mathrm{2}} } \\ $$$$\left(\mathrm{9}{D}^{\mathrm{2}} +\mathrm{6}{D}+\mathrm{1}\right){x}=−\mathrm{50}{sint} \\ $$$${x}=\frac{−\mathrm{50}}{\mathrm{9}{D}^{\mathrm{2}} +\mathrm{6}{D}+\mathrm{1}}×{sint} \\ $$$$=−\mathrm{50}×\frac{\mathrm{9}{D}^{\mathrm{2}} +\mathrm{1}−\mathrm{6}{D}}{\left(\mathrm{9}{D}^{\mathrm{2}} +\mathrm{1}\right)^{\mathrm{2}} −\mathrm{36}{D}^{\mathrm{2}} }×{sint} \\ $$$${D}\left({sint}\right)={cost}\:\:\:{D}^{\mathrm{2}} \left({sint}\right)=−{sint} \\ $$$$ \\ $$$$=−\mathrm{50}×\frac{−\mathrm{9}{sint}+{sint}−\mathrm{6}{cost}}{\left(\mathrm{9}×−\mathrm{1}^{\mathrm{2}} +\mathrm{1}\right)^{\mathrm{2}} −\mathrm{36}\left(−\mathrm{1}^{\mathrm{2}} \right)} \\ $$$$=−\mathrm{50}×\frac{−\mathrm{8}{sint}−\mathrm{6}{cost}}{\mathrm{64}+\mathrm{36}} \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}}×\left(\mathrm{8}{sint}+\mathrm{6}{cost}\right)=\mathrm{4}{sint}+\mathrm{3}{cost} \\ $$$$\mathrm{5}\left(\frac{\mathrm{4}}{\mathrm{5}\:}{sint}+\frac{\mathrm{3}}{\mathrm{5}}{cost}\right)\:\left[\:\frac{\mathrm{4}}{\mathrm{5}}={sin}\alpha\:\:\:\frac{\mathrm{3}}{\mathrm{5}}={cos}\alpha\right] \\ $$$$=\mathrm{5}{cos}\left({t}−\alpha\right) \\ $$$${complete}\:{solution}\:{is} \\ $$$${x}={e}^{\frac{−{t}}{\mathrm{3}}} \left({At}+{B}\right)+\mathrm{5}{cos}\left({t}−\alpha\right) \\ $$$$ \\ $$
Commented by peter frank last updated on 07/Dec/18
$$\mathrm{thank}\:\mathrm{you}\:\mathrm{sir} \\ $$$$ \\ $$
Commented by tanmay.chaudhury50@gmail.com last updated on 07/Dec/18
$${most}\:{welcome}… \\ $$