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Misc 2 - Verify given function is a solution of differential

Misc 2 - Chapter 9 Class 12 Differential Equations - Part 2
Misc 2 - Chapter 9 Class 12 Differential Equations - Part 3

 

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Transcript

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 ...(1) (Given equation) ...(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 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.