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Category: Differential Equation

Prove-that-for-all-values-of-u-the-point-a-cosh-u-b-sinh-u-lies-on-the-hyperbola-whose-equation-is-x-2-a-2-y-2-b-2-1-And-that-the-tangent-at-that-point-is-x-a-cosh-u-

Question Number 28468 by tawa tawa last updated on 26/Jan/18 $$\mathrm{Prove}\:\mathrm{that}\:\mathrm{for}\:\mathrm{all}\:\mathrm{values}\:\mathrm{of}\:\mathrm{u},\:\mathrm{the}\:\mathrm{point}\:\left[\mathrm{a}\:\mathrm{cosh}\left(\mathrm{u}\right)\:,\:\:\mathrm{b}\:\mathrm{sinh}\left(\mathrm{u}\right)\right]\:\mathrm{lies}\:\mathrm{on}\:\mathrm{the}\: \\ $$$$\mathrm{hyperbola}\:\mathrm{whose}\:\mathrm{equation}\:\mathrm{is}.\:\:\:\:\:\frac{\mathrm{x}^{\mathrm{2}} }{\mathrm{a}^{\mathrm{2}} }\:−\:\frac{\mathrm{y}^{\mathrm{2}} }{\mathrm{b}^{\mathrm{2}} }\:=\:\mathrm{1} \\ $$$$\mathrm{And}\:\mathrm{that}\:\mathrm{the}\:\mathrm{tangent}\:\mathrm{at}\:\mathrm{that}\:\mathrm{point}\:\mathrm{is}\::\:\:\:\frac{\mathrm{x}}{\mathrm{a}}\:\mathrm{cosh}\left(\mathrm{u}\right)\:\:−\:\:\frac{\mathrm{y}}{\mathrm{b}}\:\mathrm{sinh}\left(\mathrm{u}\right) \\ $$ Terms of Service Privacy…

xy-2-x-1-y-x-2-y-0-

Question Number 159346 by Ar Brandon last updated on 15/Nov/21 $${xy}''+\mathrm{2}\left({x}+\mathrm{1}\right){y}'+\left({x}+\mathrm{2}\right){y}=\mathrm{0} \\ $$ Answered by mindispower last updated on 16/Nov/21 $${x}\left({y}''+\mathrm{2}{y}'+{y}\right)+\left({y}'+{y}\right) \\ $$$$={x}\left(\left({y}'+{y}\right)'+\left({y}'+{y}\right)\right)+\left({y}'+{y}\right)=\mathrm{0} \\ $$$${z}={y}+{y}'…