[Class 12] If (a+bx) e^x/y = x, then prove that xy'' = (a/a+bx)^2

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

Question 31 - CBSE Class 12 Sample Paper for 2024 Boards - Solutions of Sample Papers and Past Year Papers - for Class 12 Boards

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

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