Ex 9.3, 3 - Form differential equation: y = a e3x + b e-2x

Ex 9.3, 3 - Chapter 9 Class 12 Differential Equations - Part 2

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


Ex 9.3, 3 Form a differential equation representing the given family of curves by eliminating arbitrary constants ๐‘Ž and ๐‘. ๐‘ฆ=๐‘Ž ๐‘’^3๐‘ฅ+๐‘ ๐‘’^(โˆ’2๐‘ฅ) Since it has two variables, we will differentiate twice ๐‘ฆ=๐‘Ž ๐‘’^3๐‘ฅ+๐‘ ๐‘’^(โˆ’2๐‘ฅ) โˆด Differentiating Both Sides w.r.t. ๐‘ฅ ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ=๐‘‘/๐‘‘๐‘ฅ [๐‘Ž๐‘’^3๐‘ฅ+๐‘ ๐‘’^(โˆ’2๐‘ฅ) ] =๐‘Ž๐‘’^3๐‘ฅร—3+๐‘ ๐‘’^(โˆ’2๐‘ฅ)ร—(โˆ’2) =3๐‘Ž๐‘’^3๐‘ฅโˆ’2๐‘ ๐‘’^(โˆ’2๐‘ฅ) โˆด ๐‘ฆ^โ€ฒ=3๐‘Ž๐‘’^3๐‘ฅโˆ’2๐‘ ๐‘’^(โˆ’2๐‘ฅ) ...(1) ๐‘ฆ^โ€ฒ=3๐‘Ž๐‘’^3๐‘ฅโˆ’2๐‘ ๐‘’^(โˆ’2๐‘ฅ) Again differentiating w.r.t. ๐‘ฅ ๐‘ฆ^โ€ฒโ€ฒ=๐‘‘/๐‘‘๐‘ฅ [3๐‘Ž๐‘’^3๐‘ฅโˆ’2๐‘ ๐‘’^(โˆ’2๐‘ฅ) ] ๐‘ฆ^โ€ฒโ€ฒ=3๐‘Ž๐‘’^3๐‘ฅ (3)โˆ’2๐‘ ๐‘’^(โˆ’2๐‘ฅ) (โˆ’2) โˆด ๐‘ฆ^โ€ฒโ€ฒ=9๐‘Ž๐‘’^3๐‘ฅ+4๐‘ ๐‘’^(โˆ’2๐‘ฅ) Subtracting (2) From (1) ๐‘ฆ^โ€ฒโ€ฒโˆ’๐‘ฆ^โ€ฒ=9๐‘Ž๐‘’^3๐‘ฅ+4๐‘ ๐‘’^(โˆ’2๐‘ฅ)โˆ’3๐‘Ž๐‘’^3๐‘ฅ+2๐‘ ๐‘’^(โˆ’2๐‘ฅ) ๐‘ฆ^โ€ฒโ€ฒโˆ’๐‘ฆ^โ€ฒ=6๐‘Ž๐‘’^3๐‘ฅ+6๐‘ ๐‘’^(โˆ’2๐‘ฅ) ๐‘ฆ^โ€ฒโ€ฒโˆ’๐‘ฆ^โ€ฒ=6(๐‘Ž๐‘’^3๐‘ฅ+๐‘๐‘’^(โˆ’2๐‘ฅ)) ๐‘ฆ^โ€ฒโ€ฒโˆ’๐‘ฆ^โ€ฒ=6y ๐’š^โ€ฒโ€ฒโˆ’๐’š^โ€ฒโˆ’๐Ÿ”๐’š=๐ŸŽ is the required differential equation. (As y = ๐‘Ž^3๐‘ฅ + b๐‘’^3๐‘ฅ)

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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.