Example 37 - Chapter 5 Class 12 Continuity and Differentiability

Last updated at July 26, 2026 by

Ā  Example 37 - If y = 3e2x + 2e3x, prove d2y/dx2 - 5 dy/dx - Examples

part 2 - Example 37 - Examples - Serial order wise - Chapter 5 Class 12 Continuity and Differentiability
part 3 - Example 37 - Examples - Serial order wise - Chapter 5 Class 12 Continuity and Differentiability part 4 - Example 37 - Examples - Serial order wise - Chapter 5 Class 12 Continuity and Differentiability

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Example 37 If y = 3e2x + 2e3x, prove that š‘‘2š‘¦/š‘‘š‘„2 āˆ’ 5 š‘‘š‘¦/š‘‘š‘„ + 6y = 0. Given, š‘¦ = 3š‘’2š‘„ + 2š‘’3š‘„ Differentiating š‘¤.š‘Ÿ.š‘”.š‘„ š‘‘š‘¦/š‘‘š‘„ = š‘‘(3š‘’2š‘„ + 2š‘’3š‘„)/š‘‘š‘„ š‘‘š‘¦/š‘‘š‘„ = š‘‘(3š‘’ 2š‘„)/š‘‘š‘„ + š‘‘(2š‘’ 3š‘„)/š‘‘š‘„ š‘‘š‘¦/š‘‘š‘„ = 3. š‘’2š‘„ .š‘‘(2š‘„)/š‘‘š‘„ + 2 .š‘’ 3š‘„ . š‘‘(3š‘„)/š‘‘š‘„ š‘‘š‘¦/š‘‘š‘„ = 3. š‘’2š‘„ . 2 + 2 .š‘’ 3š‘„. 3 š‘‘š‘¦/š‘‘š‘„ = 6š‘’2š‘„ + 6š‘’3š‘„ š‘‘š‘¦/š‘‘š‘„ = 6 (š‘’2š‘„ + š‘’3š‘„) Now, š’…š’š/š’…š’™ = 6 (š’†šŸš’™ + š’†šŸ‘š’™) Again Differentiating š‘¤.š‘Ÿ.š‘”.š‘„ (š‘‘^2 š‘¦)/怖š‘‘š‘„怗^2 = (š‘‘ (6(š‘’2š‘„" + " š‘’3š‘„)))/š‘‘š‘„ (š‘‘^2 š‘¦)/怖š‘‘š‘„怗^2 = 6 š‘‘(š‘’2š‘„" + " š‘’3š‘„)/š‘‘š‘„ (š‘‘^2 š‘¦)/怖š‘‘š‘„怗^2 = 6(š‘‘(š‘’2š‘„)/š‘‘š‘„ + š‘‘(š‘’3š‘„)/š‘‘š‘„) (š‘‘^2 š‘¦)/怖š‘‘š‘„怗^2 = 6(š‘’2š‘„. 2+š‘’3š‘„.3) (š’…^šŸ š’š)/怖š’…š’™ć€—^šŸ = 6(šŸš’†šŸš’™+šŸ‘š’†šŸ‘š’™) Now we need to prove š’…šŸš’š/š’…š’™šŸ āˆ’ 5 š’…š’š/š’…š’™ + 6y = 0 Solving L.H.S š‘‘2š‘¦/š‘‘š‘„2 āˆ’ 5 š‘‘š‘¦/š‘‘š‘„ + 6y = 6(2š‘’2š‘„+3š‘’3š‘„) āˆ’ 5.6 (š‘’2š‘„+š‘’3š‘„) + 6(3š‘2š‘„+2š‘’3š‘„) = 12š‘’2š‘„ + 18š‘’3š‘„ āˆ’ 30š‘’2š‘„ āˆ’ 30š‘’3š‘„ + 18š‘’2š‘„ + 12š‘’3š‘„ = 12š‘’2š‘„ āˆ’ 30š‘’2š‘„ + 18š‘’2š‘„ + 18š‘’3š‘„ āˆ’ 30š‘’3š‘„ + 12š‘’3š‘„ = 30š‘’2š‘„ āˆ’ 30š‘’2š‘„ + 30š‘’3š‘„ āˆ’ 30š‘’3š‘„ = 0 =RHS Hence proved

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