Example 40 - If y = 3e2x + 2e3x, prove d2y/dx2 - 5 dy/dx

Advertisement

Example 40 - Chapter 5 Class 12 Continuity and Differentiability - Part 2

Advertisement

Example 40 - Chapter 5 Class 12 Continuity and Differentiability - Part 3 Example 40 - Chapter 5 Class 12 Continuity and Differentiability - Part 4

  1. Chapter 5 Class 12 Continuity and Differentiability (Term 1)
  2. Serial order wise

Transcript

Example 40 If y = 3e2x + 2e3x, prove that ๐‘‘2๐‘ฆ/๐‘‘๐‘ฅ2 โˆ’ 5 ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ + 6y = 0. Given, ๐‘ฆ = 3๐‘’2๐‘ฅ + 2๐‘’3๐‘ฅ Differentiating ๐‘ค.๐‘Ÿ.๐‘ก.๐‘ฅ ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ = ๐‘‘(3๐‘’2๐‘ฅ + 2๐‘’3๐‘ฅ)/๐‘‘๐‘ฅ ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ = ๐‘‘(3๐‘’ 2๐‘ฅ)/๐‘‘๐‘ฅ + ๐‘‘(2๐‘’ 3๐‘ฅ)/๐‘‘๐‘ฅ ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ = 3. ๐‘’2๐‘ฅ .๐‘‘(2๐‘ฅ)/๐‘‘๐‘ฅ + 2 .๐‘’ 3๐‘ฅ . ๐‘‘(3๐‘ฅ)/๐‘‘๐‘ฅ ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ = 3. ๐‘’2๐‘ฅ . 2 + 2 .๐‘’ 3๐‘ฅ. 3 ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ = 6๐‘’2๐‘ฅ + 6๐‘’3๐‘ฅ ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ = 6 (๐‘’2๐‘ฅ + ๐‘’3๐‘ฅ) Now, ๐’…๐’š/๐’…๐’™ = 6 (๐’†๐Ÿ๐’™ + ๐’†๐Ÿ‘๐’™) Again Differentiating ๐‘ค.๐‘Ÿ.๐‘ก.๐‘ฅ (๐‘‘^2 ๐‘ฆ)/ใ€–๐‘‘๐‘ฅใ€—^2 = (๐‘‘ (6(๐‘’2๐‘ฅ" + " ๐‘’3๐‘ฅ)))/๐‘‘๐‘ฅ (๐‘‘^2 ๐‘ฆ)/ใ€–๐‘‘๐‘ฅใ€—^2 = 6 ๐‘‘(๐‘’2๐‘ฅ" + " ๐‘’3๐‘ฅ)/๐‘‘๐‘ฅ (๐‘‘^2 ๐‘ฆ)/ใ€–๐‘‘๐‘ฅใ€—^2 = 6(๐‘‘(๐‘’2๐‘ฅ)/๐‘‘๐‘ฅ + ๐‘‘(๐‘’3๐‘ฅ)/๐‘‘๐‘ฅ) (๐‘‘^2 ๐‘ฆ)/ใ€–๐‘‘๐‘ฅใ€—^2 = 6(๐‘’2๐‘ฅ. 2+๐‘’3๐‘ฅ.3) (๐’…^๐Ÿ ๐’š)/ใ€–๐’…๐’™ใ€—^๐Ÿ = 6(๐Ÿ๐’†๐Ÿ๐’™+๐Ÿ‘๐’†๐Ÿ‘๐’™) Now we need to prove ๐’…๐Ÿ๐’š/๐’…๐’™๐Ÿ โˆ’ 5 ๐’…๐’š/๐’…๐’™ + 6y = 0 Solving L.H.S ๐‘‘2๐‘ฆ/๐‘‘๐‘ฅ2 โˆ’ 5 ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ + 6y = 6(2๐‘’2๐‘ฅ+3๐‘’3๐‘ฅ) โˆ’ 5.6 (๐‘’2๐‘ฅ+๐‘’3๐‘ฅ) + 6(3๐‘2๐‘ฅ+2๐‘’3๐‘ฅ) = 12๐‘’2๐‘ฅ + 18๐‘’3๐‘ฅ โˆ’ 30๐‘’2๐‘ฅ โˆ’ 30๐‘’3๐‘ฅ + 18๐‘’2๐‘ฅ + 12๐‘’3๐‘ฅ = 12๐‘’2๐‘ฅ โˆ’ 30๐‘’2๐‘ฅ + 18๐‘’2๐‘ฅ + 18๐‘’3๐‘ฅ โˆ’ 30๐‘’3๐‘ฅ + 12๐‘’3๐‘ฅ = 30๐‘’2๐‘ฅ โˆ’ 30๐‘’2๐‘ฅ + 30๐‘’3๐‘ฅ โˆ’ 30๐‘’3๐‘ฅ = 0 =RHS Hence proved

About the Author

Davneet Singh's photo - Teacher, Engineer, Marketer
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.