Question 3 - Formation of Differntial equation when general solution given - Chapter 9 Class 12 Differential Equations

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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

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Question 3 Form a differential equation representing the given family of curves by eliminating arbitrary constants š‘Ž and š‘. š‘¦=š‘Ž š‘’^3š‘„+š‘ š‘’^(āˆ’2š‘„) Since it has two variables, we will differentiate twice š‘¦=š‘Ž š‘’^3š‘„+š‘ š‘’^(āˆ’2š‘„) āˆ“ Differentiating Both Sides w.r.t. š‘„ š‘‘š‘¦/š‘‘š‘„=š‘‘/š‘‘š‘„ [š‘Žš‘’^3š‘„+š‘ š‘’^(āˆ’2š‘„) ] =š‘Žš‘’^3š‘„Ɨ3+š‘ š‘’^(āˆ’2š‘„)Ɨ(āˆ’2) =3š‘Žš‘’^3š‘„āˆ’2š‘ š‘’^(āˆ’2š‘„) āˆ“ š‘¦^ā€²=3š‘Žš‘’^3š‘„āˆ’2š‘ š‘’^(āˆ’2š‘„) ...(1) š‘¦^ā€²=3š‘Žš‘’^3š‘„āˆ’2š‘ š‘’^(āˆ’2š‘„) Again differentiating w.r.t. š‘„ š‘¦^ā€²ā€²=š‘‘/š‘‘š‘„ [3š‘Žš‘’^3š‘„āˆ’2š‘ š‘’^(āˆ’2š‘„) ] š‘¦^ā€²ā€²=3š‘Žš‘’^3š‘„ (3)āˆ’2š‘ š‘’^(āˆ’2š‘„) (āˆ’2) āˆ“ š‘¦^ā€²ā€²=9š‘Žš‘’^3š‘„+4š‘ š‘’^(āˆ’2š‘„) Subtracting (2) From (1) š‘¦^ā€²ā€²āˆ’š‘¦^ā€²=9š‘Žš‘’^3š‘„+4š‘ š‘’^(āˆ’2š‘„)āˆ’3š‘Žš‘’^3š‘„+2š‘ š‘’^(āˆ’2š‘„) š‘¦^ā€²ā€²āˆ’š‘¦^ā€²=6š‘Žš‘’^3š‘„+6š‘ š‘’^(āˆ’2š‘„) š‘¦^ā€²ā€²āˆ’š‘¦^ā€²=6(š‘Žš‘’^3š‘„+š‘š‘’^(āˆ’2š‘„)) š‘¦^ā€²ā€²āˆ’š‘¦^ā€²=6y š’š^ā€²ā€²āˆ’š’š^ā€²āˆ’šŸ”š’š=šŸŽ is the required differential equation. (As y = š‘Ž^3š‘„ + bš‘’^3š‘„)

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