Question Number 54857 by Tawa1 last updated on 13/Feb/19
Answered by tanmay.chaudhury50@gmail.com last updated on 13/Feb/19
$$\frac{{d}^{\mathrm{2}} {y}}{{dx}^{\mathrm{2}} }+\lambda{y}=\mathrm{0} \\ $$$${y}={e}^{{mx}} \\ $$$${m}^{\mathrm{2}} {e}^{{mx}} +\lambda{e}^{{mx}} =\mathrm{0} \\ $$$${e}^{{mx}} \left({m}^{\mathrm{2}} +\lambda\right)=\mathrm{0} \\ $$$${e}^{{mx}} \neq\mathrm{0} \\ $$$${m}^{\mathrm{2}} =−\lambda \\ $$$${m}=\pm{i}\sqrt{\lambda}\: \\ $$$${y}={Ae}^{{i}\sqrt{\lambda}\:{x}} +{Be}^{−{i}\sqrt{\lambda}\:{x}} \:\:\:\:\left({general}\:{solution}\right) \\ $$$$\frac{{dy}}{{dx}}={A}\left({i}\sqrt{\lambda}\:\right){e}^{{i}\sqrt{\lambda}\:{x}} +{B}\left(−{i}\sqrt{\lambda}\:\right){e}^{−{i}\sqrt{\lambda}\:{x}} \\ $$$${putting}\:{condition}\rightarrow\left(\frac{{dy}}{{dx}}\right)_{{x}=\mathrm{0}} +\left({y}\right)_{{x}=\mathrm{0}} =\mathrm{0} \\ $$$${we}\:{get} \\ $$$${A}\left({i}\sqrt{\lambda}\:\right)+{B}\left(−{i}\sqrt{\lambda}\:\right)+{A}+{B}=\mathrm{0} \\ $$$$\left({A}+{B}\right)+{i}\sqrt{\lambda}\:\left({A}−{B}\right)=\mathrm{0} \\ $$$$ \\ $$$${A}+{B}=\mathrm{0}\:\:{A}−{B}=\mathrm{0}\:{so}\:{A}=\mathrm{0}\:\:{B}=\mathrm{0} \\ $$$${peculiar}\:{conclusion}\:{is}\:{coming}… \\ $$$${but}\:{both}\:{A}\neq\mathrm{0}\:\:\:{B}\neq\mathrm{0} \\ $$$${something}\:{wrong}…{condition}\:\mathrm{1} \\ $$$${as}\:{per}\:{condition}\:\mathrm{2} \\ $$$${Ae}^{{i}\sqrt{\lambda}\:} +{Be}^{−{i}\sqrt{\lambda}\:} +\mathrm{3}\left\{{A}\left({i}\sqrt{\lambda}\right){e}^{{i}\sqrt{\lambda}} +{B}\left(−{i}\sqrt{\lambda}\:\right){e}^{−{i}\sqrt{\left(\lambda\right.}\:} =\mathrm{0}\right. \\ $$$${Ae}^{{i}\sqrt{\lambda}} +{Be}^{−{i}\sqrt{\lambda}\:} =\mathrm{0} \\ $$$$\mathrm{3}{i}\sqrt{\lambda}\:\left({Ae}^{{i}\sqrt{\lambda}\:} −{Be}^{−{i}\sqrt{\lambda}\:} \right)=\mathrm{0} \\ $$$${Ae}^{{i}\sqrt{\lambda}\:} ={Be}^{−{i}\sqrt{\lambda}\:} \\ $$$${same}\:{peculiar}\:{conclusioncoming}… \\ $$$${let}\:{other}\:{check}…. \\ $$
Commented by Tawa1 last updated on 13/Feb/19
$$\mathrm{God}\:\mathrm{bless}\:\mathrm{you}\:\mathrm{sir}.\:\mathrm{Waiting} \\ $$