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

 

  1. Chapter 9 Class 12 Differential Equations (Term 2)
  2. Serial order wise

Transcript

Misc 2 For each of the exercise given below , verify that the given function (๐‘–๐‘š๐‘๐‘™๐‘–๐‘๐‘–๐‘ก ๐‘œ๐‘Ÿ ๐‘’๐‘ฅ๐‘๐‘™๐‘–๐‘๐‘–๐‘ก) is a solution of the corresponding differential equation . (i) ๐‘ฅ๐‘ฆ=๐‘Ž ๐‘’^๐‘ฅ+๐‘ ๐‘’^(โˆ’๐‘ฅ)+๐‘ฅ^2 : ๐‘ฅ (๐‘‘^2 ๐‘ฆ)/(๐‘‘๐‘ฅ^2 )+2 ๐‘‘๐‘ฆ/๐‘‘๐‘ฅโˆ’๐‘ฅ๐‘ฆ+๐‘ฅ^2โˆ’2=0 ๐‘ฅ๐‘ฆ=๐‘Ž๐‘’^๐‘ฅ+๐‘ ๐‘’^(โˆ’๐‘ฅ)+๐‘ฅ^2 Differentiating w.r.t x (๐‘‘(๐‘ฅ๐‘ฆ))/๐‘‘๐‘ฅ=๐‘‘/๐‘‘๐‘ฅ [๐‘Ž ๐‘’^๐‘ฅ+๐‘ ๐‘’^(โˆ’๐‘ฅ)+๐‘ฅ^2 ] ๐‘‘๐‘ฅ/๐‘‘๐‘ฅ y+๐‘‘๐‘ฆ/๐‘‘๐‘ฅ ๐‘ฅ =๐‘Žใ€– ๐‘’ใ€—^๐‘ฅ+(โˆ’1)๐‘ ๐‘’^(โˆ’๐‘ฅ)+2๐‘ฅ y+y^โ€ฒ x =aใ€– eใ€—^xโˆ’b e^(โˆ’x)+2x Differentiating again w.r.t x ๐‘ฆโ€ฒ+(๐‘ฆ^โ€ฒ ๐‘ฅ)^โ€ฒ =(๐‘Ž๐‘’^๐‘ฅ )^โ€ฒโˆ’(๐‘๐‘’^(โˆ’๐‘ฅ) )^โ€ฒ+(2๐‘ฅ)^โ€ฒ ๐‘ฆ^โ€ฒ+(๐‘ฆ^โ€ฒโ€ฒ ๐‘ฅ+๐‘ฆ^โ€ฒร—1)=๐‘Ž๐‘’^๐‘ฅ+๐‘๐‘’^(โˆ’๐‘ฅ)+2 ๐‘ฆ^โ€ฒโ€ฒ ๐‘ฅ+2๐‘ฆ^โ€ฒ=๐‘Ž๐‘’^๐‘ฅ+๐‘๐‘’^(โˆ’๐‘ฅ)+2 Now, we know that ๐‘ฅ๐‘ฆ=๐‘Ž๐‘’^๐‘ฅ+๐‘ ๐‘’^(โˆ’๐‘ฅ)+๐‘ฅ^2 ๐‘ฅ๐‘ฆโˆ’๐‘ฅ^2=๐‘Ž๐‘’^๐‘ฅ+๐‘ ๐‘’^(โˆ’๐‘ฅ) ๐‘Ž๐‘’^๐‘ฅ+๐‘ ๐‘’^(โˆ’๐‘ฅ)=๐‘ฅ๐‘ฆโˆ’๐‘ฅ^2 ...(1) (Given equation) ...(2) Putting (2) in (1) ๐‘ฆ^โ€ฒโ€ฒ ๐‘ฅ+2๐‘ฆ^โ€ฒ=๐’‚๐’†^๐’™+๐’ƒ๐’†^(โˆ’๐’™)+2 ๐‘ฆ^โ€ฒโ€ฒ ๐‘ฅ+2๐‘ฆ^โ€ฒ=๐’™๐’šโˆ’๐’™^๐Ÿ+2 (๐‘‘^2 ๐‘ฆ)/(๐‘‘๐‘ฅ^2 ) ๐‘ฅ+2 ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ=๐’™๐’šโˆ’๐’™^๐Ÿ+2 (๐’…^๐Ÿ ๐’š)/(๐’…๐’™^๐Ÿ ) ๐’™+๐Ÿ ๐’…๐’š/๐’…๐’™โˆ’๐’™๐’š+๐’™^๐Ÿ=๐ŸŽ โˆด The given function is a solution

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Davneet Singh
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 10 years. He provides courses for Maths and Science at Teachoo.