Question Number 204418 by peter frank last updated on 17/Feb/24
Commented by mr W last updated on 19/Feb/24
$${y}=\mathrm{2}\left({x}−\frac{\mathrm{1}}{{x}}\right)^{\mathrm{2}} \\ $$
Answered by mr W last updated on 19/Feb/24
$${x}={e}^{{t}} \\ $$$$\frac{{dx}}{{dt}}={e}^{{t}} ={x} \\ $$$$\frac{{dy}}{{dx}}=\frac{{dy}}{{dt}}×\frac{\mathrm{1}}{\frac{{dx}}{{dt}}}=\frac{{dy}}{{dt}}×\frac{\mathrm{1}}{{x}} \\ $$$$\Rightarrow{x}\frac{{dy}}{{dx}}=\frac{{dy}}{{dt}} \\ $$$$\frac{{d}^{\mathrm{2}} {y}}{{dx}^{\mathrm{2}} }=\frac{{d}}{{dx}}\left(\frac{{dy}}{{dx}}\right)=\frac{{d}}{{dt}}\left(\frac{{dy}}{{dx}}\right)×\frac{\mathrm{1}}{\frac{{dx}}{{dt}}}=\frac{{d}}{{dt}}\left(\frac{{dy}}{{dt}}×\frac{\mathrm{1}}{{x}}\right)×\frac{\mathrm{1}}{{x}} \\ $$$$=\left(\frac{{d}^{\mathrm{2}} {y}}{{dt}^{\mathrm{2}} }×\frac{\mathrm{1}}{{x}}+\frac{{dy}}{{dt}}×\left(−\frac{\mathrm{1}}{{x}^{\mathrm{2}} }×\frac{{dx}}{{dt}}\right)\right)×\frac{\mathrm{1}}{{x}} \\ $$$$=\left(\frac{{d}^{\mathrm{2}} {y}}{{dt}^{\mathrm{2}} }×\frac{\mathrm{1}}{{x}}+\frac{{dy}}{{dt}}×\left(−\frac{\mathrm{1}}{{x}^{\mathrm{2}} }×{x}\right)\right)×\frac{\mathrm{1}}{{x}} \\ $$$$=\left(\frac{{d}^{\mathrm{2}} {y}}{{dt}^{\mathrm{2}} }−\frac{{dy}}{{dt}}\right)×\frac{\mathrm{1}}{{x}^{\mathrm{2}} } \\ $$$$\Rightarrow{x}^{\mathrm{2}} \frac{{d}^{\mathrm{2}} {y}}{{dx}^{\mathrm{2}} }=\frac{{d}^{\mathrm{2}} {y}}{{dt}^{\mathrm{2}} }−\frac{{dy}}{{dt}} \\ $$$${x}^{\mathrm{2}} \frac{{d}^{\mathrm{2}} {y}}{{dx}^{\mathrm{2}} }+{x}\frac{{dy}}{{dx}}−\mathrm{4}{y}=\mathrm{16} \\ $$$$\frac{{d}^{\mathrm{2}} {y}}{{dt}^{\mathrm{2}} }−\frac{{dy}}{{dt}}+\frac{{dy}}{{dt}}−\mathrm{4}{y}=\mathrm{16} \\ $$$$\frac{{d}^{\mathrm{2}} {y}}{{dt}^{\mathrm{2}} }−\mathrm{4}{y}=\mathrm{16} \\ $$$$\frac{{d}^{\mathrm{2}} \left({y}+\mathrm{4}\right)}{{dt}^{\mathrm{2}} }−\mathrm{4}\left({y}+\mathrm{4}\right)=\mathrm{0} \\ $$$$\Rightarrow{y}+\mathrm{4}={A}\:{e}^{\mathrm{2}{t}} +{B}\:{e}^{−\mathrm{2}{t}} \\ $$$${y}\left({x}=\mathrm{1}\right)=\mathrm{0}\:\Rightarrow{y}\left({t}=\mathrm{0}\right)=\mathrm{0}\:\Rightarrow \\ $$$$\mathrm{0}+\mathrm{4}={A}+{B} \\ $$$$\frac{{dy}}{{dx}}\mid_{{x}=\mathrm{1}} =\mathrm{0}\:\Rightarrow\frac{{dy}}{{dt}}\mid_{{t}=\mathrm{0}} =\mathrm{0}\:\Rightarrow \\ $$$$\mathrm{0}=\mathrm{2}{A}−\mathrm{2}{B} \\ $$$$\Rightarrow{A}={B}=\mathrm{2} \\ $$$$\Rightarrow{y}+\mathrm{4}=\mathrm{2}\left({e}^{\mathrm{2}{t}} +{e}^{−\mathrm{2}{t}} \right)=\mathrm{2}\left({x}^{\mathrm{2}} +\frac{\mathrm{1}}{{x}^{\mathrm{2}} }\right) \\ $$$$\Rightarrow{y}=\mathrm{2}\left({x}^{\mathrm{2}} +\frac{\mathrm{1}}{{x}^{\mathrm{2}} }\right)−\mathrm{4}=\mathrm{2}\left({x}−\frac{\mathrm{1}}{{x}}\right)^{\mathrm{2}} \\ $$
Commented by peter frank last updated on 21/Feb/24
$$\mathrm{thank}\:\mathrm{you} \\ $$