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Misc 2 For each of the exercise given below , verify that the given function (𝑖𝑚𝑝𝑙𝑖𝑐𝑖𝑡 𝑜𝑟 𝑒𝑥𝑝𝑙𝑖𝑐𝑖𝑡) is a solution of the corresponding differential equation . (i) 𝑥𝑦=𝑎 𝑒^𝑥+𝑏 𝑒^(−𝑥)+𝑥^2 : 𝑥 (𝑑^2 𝑦)/(𝑑𝑥^2 )+2 𝑑𝑦/𝑑𝑥−𝑥𝑦+𝑥^2−2=0 𝑥𝑦=𝑎𝑒^𝑥+𝑏 𝑒^(−𝑥)+𝑥^2 Differentiating w.r.t x (𝑑(𝑥𝑦))/𝑑𝑥=𝑑/𝑑𝑥 [𝑎 𝑒^𝑥+𝑏 𝑒^(−𝑥)+𝑥^2 ] 𝑑𝑥/𝑑𝑥 y+𝑑𝑦/𝑑𝑥 𝑥 =𝑎〖 𝑒〗^𝑥+(−1)𝑏 𝑒^(−𝑥)+2𝑥 𝒚+𝒚^′ 𝒙 =𝒂〖 𝒆〗^𝒙−𝒃 𝒆^(−𝒙)+𝟐𝒙 Differentiating again w.r.t x 𝑦′+(𝑦^′ 𝑥)^′ =(𝑎𝑒^𝑥 )^′−(𝑏𝑒^(−𝑥) )^′+(2𝑥)^′ 𝑦^′+(𝑦^′′ 𝑥+𝑦^′×1)=𝑎𝑒^𝑥+𝑏𝑒^(−𝑥)+2 𝒚^′′ 𝒙+𝟐𝒚^′=𝒂𝒆^𝒙+𝒃𝒆^(−𝒙)+𝟐 Now, we know that 𝑥𝑦=𝑎𝑒^𝑥+𝑏 𝑒^(−𝑥)+𝑥^2 𝑥𝑦−𝑥^2=𝑎𝑒^𝑥+𝑏 𝑒^(−𝑥) 𝒂𝒆^𝒙+𝒃 𝒆^(−𝒙)=𝒙𝒚−𝒙^𝟐 Putting (2) in (1) 𝑦^′′ 𝑥+2𝑦^′=𝒂𝒆^𝒙+𝒃𝒆^(−𝒙)+2 𝑦^′′ 𝑥+2𝑦^′=𝒙𝒚−𝒙^𝟐+2 (𝑑^2 𝑦)/(𝑑𝑥^2 ) 𝑥+2 𝑑𝑦/𝑑𝑥=𝒙𝒚−𝒙^𝟐+2 (𝒅^𝟐 𝒚)/(𝒅𝒙^𝟐 ) 𝒙+𝟐 𝒅𝒚/𝒅𝒙−𝒙𝒚+𝒙^𝟐=𝟎 ∴ The given function is a solution

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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 and Computer Science at Teachoo