If (a+bx)e^(y/x)=x then prove that x (d^2 y)/(dx^2 )=(a/(a+bx))^2

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Given (𝑎+𝑏𝑥)𝑒^(𝑦/𝑥)=𝑥 𝒆^(𝒚/𝒙) = 𝒙/((𝒂 + 𝒃𝒙)) Taking log on both sides log 𝑒^(𝑦/𝑥) = log 𝑥/((𝑎 + 𝑏𝑥)) 𝒚/𝒙 𝐥𝐨𝐠 𝐞 = log x – log (a + bx) 𝑦/𝑥 × 1= log x – log (a + bx) 𝒚/𝒙 = log x – log (a + bx) Differentiating w.r.t x 𝑑(𝑦/𝑥)/𝑑𝑥 = 1/𝑥 – 1/(𝑎 + 𝑏𝑥) × 𝑏 (𝒅𝒚/𝒅𝒙 𝐱 − 𝒚)/𝒙^𝟐 = 1/𝑥 – 𝑏/(𝑎 + 𝑏𝑥) (𝑦^′ x−y)/𝑥^2 = (𝑎 + 𝑏𝑥 − 𝑏𝑥)/(𝑥(𝑎 + 𝑏𝑥)) 𝑦^′ x−y = 〖𝑎𝑥〗^2/(𝑥(𝑎+𝑏𝑥)) 𝒚^′ 𝐱−𝐲 = 𝒂𝒙/(𝒂+𝒃𝒙) Differentiating again w.r.t x (𝐝(𝒚^′ )/𝒅𝒙 𝒙+𝒚^′ 𝒅𝒙/𝒅𝒙) − 𝒅𝒚/𝒅𝒙 = (𝒂( 𝒂 + 𝒃𝒙) − 𝒃(𝒂𝒙))/〖(𝒂 + 𝒃𝒙)〗^𝟐 𝑦^′′ 𝑥+𝑦^′−𝑦^′= (𝑎^2 + 𝑏𝑎𝑥 − 𝑏𝑎𝑥)/〖(𝑎 + 𝑏𝑥)〗^2 𝑦^′′ 𝑥 = 𝑎^2/〖(𝑎 + 𝑏𝑥)〗^2 𝒙 (𝒅^𝟐 𝒚)/(𝒅𝒙^𝟐 )=(𝒂/(𝒂 + 𝒃𝒙))^𝟐 Hence proved

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Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 14 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.