Question Number 49357 by peter frank last updated on 06/Dec/18
Commented by maxmathsup by imad last updated on 01/Jan/19
$$\left.\mathrm{2}\right)\:{let}\:{prove}\:{that}\:{y}^{\left({n}\right)} \left({x}\right)=\frac{\left(−\mathrm{1}\right)^{{n}−\mathrm{1}} \left({n}−\mathrm{1}\right)!}{\left(\mathrm{1}+{x}\right)^{{n}} }\:\:{with}\:{y}\left({x}\right)={ln}\left(\mathrm{1}+{x}\right) \\ $$$${by}\:{recurrence}\:{on}\:{n}\:\:{for}\:{n}=\mathrm{1}\:\:\:{y}^{\left(\mathrm{1}\right)} \left({x}\right)=\frac{\mathrm{1}}{\mathrm{1}+{x}}\:=\:\frac{\left(−\mathrm{1}\right)^{\mathrm{1}−\mathrm{1}} \mathrm{0}!}{\left(\mathrm{1}+{x}\right)^{\mathrm{1}} }\:\:\left({true}\right)\:{let}\:{suppose} \\ $$$${y}^{\left({n}\right)} \left({x}\right)\:=\frac{\left(−\mathrm{1}\right)^{{n}−\mathrm{1}} \left({n}−\mathrm{1}\right)!}{\left(\mathrm{1}+{x}\right)^{{n}} }\:\:{we}\:{have} \\ $$$${y}^{\left({n}+\mathrm{1}\right)} \left({x}\right)\:=\frac{{d}}{{dx}}\left({y}^{\left({n}\right)} \left({x}\right)\right)=\left(−\mathrm{1}\right)^{{n}−\mathrm{1}} \left({n}−\mathrm{1}\right)!×\frac{−{n}\left(\mathrm{1}+{x}\right)^{{n}−\mathrm{1}} }{\left(\mathrm{1}+{x}\right)^{\mathrm{2}{n}} } \\ $$$$=\left(−\mathrm{1}\right)^{{n}} {n}!\:\:\:\frac{\mathrm{1}}{\left(\mathrm{1}+{x}\right)^{{n}+\mathrm{1}} }\:=\frac{\left(−\mathrm{1}\right)^{{n}+\mathrm{1}−\mathrm{1}} \left({n}+\mathrm{1}−\mathrm{1}\right)!}{\left(\mathrm{1}+{x}\right)^{{n}+\mathrm{1}} }\:\:{so}\:{the}\:{result}\:{is}\:{true}\:{at}\:{term} \\ $$$$\left({n}+\mathrm{1}\right). \\ $$
Answered by tanmay.chaudhury50@gmail.com last updated on 06/Dec/18
$$\left.\mathrm{1}\right)\frac{{d}^{\mathrm{2}} {x}}{{dt}^{\mathrm{2}} }+{k}\frac{{dx}}{{dt}}+{n}^{\mathrm{2}} {x}={pcoswt} \\ $$$${step}\mathrm{1} \\ $$$$\frac{{d}^{\mathrm{2}} {x}}{{dt}^{\mathrm{2}} }+{k}\frac{{dx}}{{dt}}+{n}^{\mathrm{2}} {x}=\mathrm{0} \\ $$$${let}\:{x}={e}^{\alpha{t}} \:\:{so}\:\frac{{dx}}{{dt}}=\alpha{e}^{\alpha{t}} \:\:\:{and}\:\:\frac{{d}^{\mathrm{2}} {x}}{{dt}^{\mathrm{2}} }=\alpha^{\mathrm{2}} {e}^{\alpha{t}} \\ $$$${e}^{\alpha{t}} \left(\alpha^{\mathrm{2}} +{k}\alpha+{n}^{\mathrm{2}} \right)=\mathrm{0}\:\:{e}^{\alpha{t}} \neq\mathrm{0} \\ $$$$\alpha^{\mathrm{2}} +{k}\alpha+{n}^{\mathrm{2}} =\mathrm{0} \\ $$$$\alpha=\frac{−{k}\pm\sqrt{{k}^{\mathrm{2}} −\mathrm{4}{n}^{\mathrm{2}} }}{\mathrm{2}} \\ $$$${Complementary}\:{function} \\ $$$${x}={Ae}^{\left(\frac{−{k}+\sqrt{{k}^{\mathrm{2}} −\mathrm{4}{n}^{\mathrm{2}} }}{\mathrm{2}}\right){t}} +{Be}^{\left(\frac{−{k}−\sqrt{{k}^{\mathrm{2}} −\mathrm{4}{n}^{\mathrm{2}} }\:}{\mathrm{2}}\right)^{} {t}} \\ $$$${step}\:\mathrm{2} \\ $$$${determination}\:{of}\:{particular}\:{intregal} \\ $$$$\frac{{d}^{\mathrm{2}} {x}}{{dt}^{\mathrm{2}} }+{k}\frac{{dx}}{{dt}}+{n}^{\mathrm{2}} {x}={pcoswt} \\ $$$${let}\:\frac{{d}}{{dt}}={D}\:\:{and}\:\frac{{d}^{\mathrm{2}} }{{dx}^{\mathrm{2}} }={D}^{\mathrm{2}} \\ $$$${so}\:\left({D}^{\mathrm{2}} +{kD}+{n}^{\mathrm{2}} \right){x}={pcoswt} \\ $$$${x}=\frac{\mathrm{1}}{{D}^{\mathrm{2}} +{n}^{\mathrm{2}} +{kD}}×{pcoswt} \\ $$$$=\frac{\left({D}^{\mathrm{2}} +{n}^{\mathrm{2}} −{kD}\right)}{\left({D}^{\mathrm{2}} +{n}^{\mathrm{2}} \right)^{\mathrm{2}} −{k}^{\mathrm{2}} {D}^{\mathrm{2}} }{pcoswt} \\ $$$${D}\left({pcoswt}\right)=−{wpsinwt} \\ $$$${D}^{\mathrm{2}} \left({pcoswt}\right)=−{w}^{\mathrm{2}} {pcoswt} \\ $$$$=\frac{\left(−{w}^{\mathrm{2}} {pcoswt}+{n}^{\mathrm{2}} {pcoswt}+{kpwsinwt}\right)}{\left(−{w}^{\mathrm{2}} +{n}^{\mathrm{2}} \right)^{\mathrm{2}} −{k}^{\mathrm{2}} \left(−{w}^{\mathrm{2}} \right)} \\ $$$$={p}×\left[\frac{\left({n}^{\mathrm{2}} −{w}^{\mathrm{2}} \right){coswt}+{kwsinwt}}{\left({n}^{\mathrm{2}} −{w}^{\mathrm{2}} \right)^{\mathrm{2}} +{k}^{\mathrm{2}} {w}^{\mathrm{2}} }\right] \\ $$$$ \\ $$$$ \\ $$$$ \\ $$$$ \\ $$
Answered by tanmay.chaudhury50@gmail.com last updated on 06/Dec/18
$$\left.\mathrm{2}\right){y}={ln}\left(\mathrm{1}+{x}\right) \\ $$$$\frac{{dy}}{{dx}}=\frac{\mathrm{1}}{\left(\mathrm{1}+{x}\right)}\:\:\:\:\frac{{d}^{\mathrm{2}} {y}}{{dx}^{\mathrm{2}} }=\frac{−\mathrm{1}}{\left(\mathrm{1}+{x}\right)^{\mathrm{2}} } \\ $$$${so}\:\:{when}\:{n}=\mathrm{2} \\ $$$${LHS}=\frac{−\mathrm{1}}{\left(\mathrm{1}+{x}\right)^{\mathrm{2}} }\:\:\:{and}\:{RHS}=\frac{\left(−\mathrm{1}\right)^{\mathrm{2}−\mathrm{1}} \left(\mathrm{2}−\mathrm{1}\right)!}{\left(\mathrm{1}+{x}\right)^{\mathrm{2}} }=\frac{−\mathrm{1}}{\left(\mathrm{1}+{x}\right)^{\mathrm{2}} } \\ $$$${so}\:{for}\:{n}=\mathrm{2}\:{the}\:{statement}\:{is}\:{true}\: \\ $$$${now}\:{let}\:{assume}\:{for}\:{n}={k}\:{the}\:{statement}\:{is}\:{true} \\ $$$${we}\:{have}\:{to}\:{prove}\:{that}\:{it}\:{is}\:{true}\:{for}\:{n}={k}+\mathrm{1} \\ $$$$\frac{{d}^{{k}} {y}}{{dx}^{{k}} }=\frac{\left(−\mathrm{1}\right)^{{k}−\mathrm{1}} \left({k}−\mathrm{1}\right)!}{\left(\mathrm{1}+{x}\right)^{{k}} } \\ $$$${so}\:\frac{{d}}{{dx}}\left(\frac{{d}^{{k}} {y}}{{dx}^{{k}} }\right)=\frac{{d}^{{k}+\mathrm{1}} {y}}{{dx}^{{k}+\mathrm{1}} }\leftarrow{LHS} \\ $$$$\frac{{d}}{{dx}}\left[\frac{\left(−\mathrm{1}\right)^{{k}−\mathrm{1}} \left({k}−\mathrm{1}\right)!}{\left(\mathrm{1}+{x}\right)^{{k}} }\right] \\ $$$$=\frac{\left(−\mathrm{1}\right)^{{k}−\mathrm{1}} \left({k}−\mathrm{1}\right)!}{\mathrm{1}}×\frac{{d}}{{dx}}\left\{\frac{\mathrm{1}}{\left(\mathrm{1}+{x}\right)^{{k}} }\right\} \\ $$$$=\left(−\mathrm{1}\right)^{{k}−\mathrm{1}} \left({k}−\mathrm{1}\right)!×\left(−{k}\right)×\left(\mathrm{1}+{x}\right)^{−{k}−\mathrm{1}} \\ $$$$=\left(−\mathrm{1}\right)^{{k}−\mathrm{1}} \left({k}−\mathrm{1}\right)!\left(−\mathrm{1}\right){k}×\frac{\mathrm{1}}{\left(\mathrm{1}+{x}\right)^{{k}+\mathrm{1}} } \\ $$$$=\left(−\mathrm{1}\right)^{{k}−\mathrm{1}+\mathrm{1}} ×{k}\left({k}−\mathrm{1}\right)!×\frac{\mathrm{1}}{\left(\mathrm{1}+{x}\right)^{{k}+\mathrm{1}} } \\ $$$$=\frac{\left(−\mathrm{1}\right)^{{k}} ×{k}!}{\left(\mathrm{1}+{x}\right)^{{k}+\mathrm{1}} }\leftarrow{this}\:{expression}\:{is}\:{same}\:{if}\:{you} \\ $$$${put}\:{n}={k}+\mathrm{1}\:{in}\:{expression} \\ $$$$\:\frac{\left(−\mathrm{1}\right)^{{n}−\mathrm{1}} ×\left({n}−\mathrm{1}\right)!}{\left(\mathrm{1}+{x}\right)^{{n}} } \\ $$$${hence}\:{proved}… \\ $$