Question Number 1964 by 123456 last updated on 26/Oct/15
$$\frac{{d}^{\mathrm{2}} {y}}{{dx}^{\mathrm{2}} }=\frac{{cy}}{{x}^{\mathrm{2}} } \\ $$$${y}\left({x}\right)=? \\ $$
Answered by prakash jain last updated on 26/Oct/15
$${y}={kx}^{{n}} \\ $$$${y}''={kn}\left({n}−\mathrm{1}\right){x}^{{n}−\mathrm{2}} ={ckx}^{{n}−\mathrm{2}} \\ $$$${n}\left({n}−\mathrm{1}\right)={c} \\ $$$${n}^{\mathrm{2}} −{n}−{c}=\mathrm{0} \\ $$$${n}=\frac{\mathrm{1}\pm\sqrt{\mathrm{1}+\mathrm{4c}}}{\mathrm{2}} \\ $$$${y}={c}_{\mathrm{1}} {x}^{\frac{\mathrm{1}+\sqrt{\mathrm{1}+\mathrm{4}{c}}}{\mathrm{2}}} +{c}_{\mathrm{2}} {x}^{\frac{\mathrm{1}−\sqrt{\mathrm{1}+\mathrm{4}{c}}}{\mathrm{2}}} \\ $$