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

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  1. Chapter 5 Class 12 Continuity and Differentiability
  2. Serial order wise
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Example 40 If y = 3e2x + 2e3x, prove that ๐‘‘2๐‘ฆ๏ทฎ๐‘‘๐‘ฅ2๏ทฏ โˆ’ 5 ๐‘‘๐‘ฆ๏ทฎ๐‘‘๐‘ฅ๏ทฏ + 6y = 0. Given, ๐‘ฆ = 3๐‘’2๐‘ฅ + 2๐‘’3๐‘ฅ Differentiating ๐‘ค.๐‘Ÿ.๐‘ก.๐‘ฅ ๐‘‘๐‘ฆ๏ทฎ๐‘‘๐‘ฅ๏ทฏ = ๐‘‘ 3๐‘’2๐‘ฅ + 2๐‘’3๐‘ฅ๏ทฏ๏ทฎ๐‘‘๐‘ฅ๏ทฏ ๐‘‘๐‘ฆ๏ทฎ๐‘‘๐‘ฅ๏ทฏ = ๐‘‘ 3๐‘’ 2๐‘ฅ๏ทฏ๏ทฎ๐‘‘๐‘ฅ๏ทฏ + ๐‘‘ 2๐‘’ 3๐‘ฅ๏ทฏ๏ทฎ๐‘‘๐‘ฅ๏ทฏ ๐‘‘๐‘ฆ๏ทฎ๐‘‘๐‘ฅ๏ทฏ = 3. ๐‘’2๐‘ฅ . ๐‘‘ 2๐‘ฅ๏ทฏ๏ทฎ๐‘‘๐‘ฅ๏ทฏ + 2 .๐‘’ 3๐‘ฅ . ๐‘‘ 3๐‘ฅ๏ทฏ๏ทฎ๐‘‘๐‘ฅ๏ทฏ ๐‘‘๐‘ฆ๏ทฎ๐‘‘๐‘ฅ๏ทฏ = 3. ๐‘’2๐‘ฅ . 2 + 2 .๐‘’ 3๐‘ฅ. 3 ๐‘‘๐‘ฆ๏ทฎ๐‘‘๐‘ฅ๏ทฏ = 6๐‘’2๐‘ฅ + 6๐‘’3๐‘ฅ ๐‘‘๐‘ฆ๏ทฎ๐‘‘๐‘ฅ๏ทฏ = 6 (๐‘’2๐‘ฅ + ๐‘’3๐‘ฅ) Now, ๐‘‘๐‘ฆ๏ทฎ๐‘‘๐‘ฅ๏ทฏ = 6 (๐‘’2๐‘ฅ + ๐‘’3๐‘ฅ) Again Differentiating ๐‘ค.๐‘Ÿ.๐‘ก.๐‘ฅ ๐‘‘๏ทฎ2๏ทฏ๐‘ฆ๏ทฎ ๐‘‘๐‘ฅ๏ทฎ2๏ทฏ๏ทฏ = ๐‘‘ 6 ๐‘’2๐‘ฅ + ๐‘’3๐‘ฅ๏ทฏ๏ทฏ๏ทฎ๐‘‘๐‘ฅ๏ทฏ ๐‘‘๏ทฎ2๏ทฏ๐‘ฆ๏ทฎ ๐‘‘๐‘ฅ๏ทฎ2๏ทฏ๏ทฏ = 6 ๐‘‘ ๐‘’2๐‘ฅ + ๐‘’3๐‘ฅ๏ทฏ๏ทฎ๐‘‘๐‘ฅ๏ทฏ ๐‘‘๏ทฎ2๏ทฏ๐‘ฆ๏ทฎ ๐‘‘๐‘ฅ๏ทฎ2๏ทฏ๏ทฏ = 6 ๐‘‘ ๐‘’2๐‘ฅ๏ทฏ๏ทฎ๐‘‘๐‘ฅ๏ทฏ + ๐‘‘ ๐‘’3๐‘ฅ๏ทฏ๏ทฎ๐‘‘๐‘ฅ๏ทฏ๏ทฏ ๐‘‘๏ทฎ2๏ทฏ๐‘ฆ๏ทฎ ๐‘‘๐‘ฅ๏ทฎ2๏ทฏ๏ทฏ = 6 ๐‘’2๐‘ฅ. 2+๐‘’3๐‘ฅ.3๏ทฏ ๐’…๏ทฎ๐Ÿ๏ทฏ๐’š๏ทฎ ๐’…๐’™๏ทฎ๐Ÿ๏ทฏ๏ทฏ = 6 ๐Ÿ๐’†๐Ÿ๐’™+๐Ÿ‘๐’†๐Ÿ‘๐’™๏ทฏ Now we need to prove ๐‘‘2๐‘ฆ๏ทฎ๐‘‘๐‘ฅ2๏ทฏ โˆ’ 5 ๐‘‘๐‘ฆ๏ทฎ๐‘‘๐‘ฅ๏ทฏ + 6y = 0 Solving L.H.S ๐‘‘2๐‘ฆ๏ทฎ๐‘‘๐‘ฅ2๏ทฏ โˆ’ 5 ๐‘‘๐‘ฆ๏ทฎ๐‘‘๐‘ฅ๏ทฏ + 6y = 6 2๐‘’2๐‘ฅ+3๐‘’3๐‘ฅ๏ทฏ โˆ’ 5.6 ๐‘’2๐‘ฅ+๐‘’3๐‘ฅ๏ทฏ + 6 3๐‘2๐‘ฅ+2๐‘’3๐‘ฅ๏ทฏ = 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

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Davneet Singh
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 8 years. He provides courses for Maths and Science at Teachoo. You can check his NCERT Solutions from Class 6 to 12.