
Ex 9.3
Ex 9.3, 2 Deleted for CBSE Board 2022 Exams
Ex 9.3, 3 Important Deleted for CBSE Board 2022 Exams You are here
Ex 9.3, 4 Deleted for CBSE Board 2022 Exams
Ex 9.3, 5 Important Deleted for CBSE Board 2022 Exams
Ex 9.3, 6 Deleted for CBSE Board 2022 Exams
Ex 9.3, 7 Important Deleted for CBSE Board 2022 Exams
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Ex 9.3, 10 Important Deleted for CBSE Board 2022 Exams
Ex 9.3, 11 (MCQ) Deleted for CBSE Board 2022 Exams
Ex 9.3, 12 (MCQ) Important Deleted for CBSE Board 2022 Exams
Last updated at Dec. 10, 2019 by Teachoo
Ex 9.3, 3 Form a differential equation representing the given family of curves by eliminating arbitrary constants 𝑎 and 𝑏. 𝑦=𝑎 𝑒^3𝑥+𝑏 𝑒^(−2𝑥) Since it has two variables, we will differentiate twice 𝑦=𝑎 𝑒^3𝑥+𝑏 𝑒^(−2𝑥) ∴ Differentiating Both Sides w.r.t. 𝑥 𝑑𝑦/𝑑𝑥=𝑑/𝑑𝑥 [𝑎𝑒^3𝑥+𝑏 𝑒^(−2𝑥) ] =𝑎𝑒^3𝑥×3+𝑏 𝑒^(−2𝑥)×(−2) =3𝑎𝑒^3𝑥−2𝑏 𝑒^(−2𝑥) ∴ 𝑦^′=3𝑎𝑒^3𝑥−2𝑏 𝑒^(−2𝑥) ...(1) 𝑦^′=3𝑎𝑒^3𝑥−2𝑏 𝑒^(−2𝑥) Again differentiating w.r.t. 𝑥 𝑦^′′=𝑑/𝑑𝑥 [3𝑎𝑒^3𝑥−2𝑏 𝑒^(−2𝑥) ] 𝑦^′′=3𝑎𝑒^3𝑥 (3)−2𝑏 𝑒^(−2𝑥) (−2) ∴ 𝑦^′′=9𝑎𝑒^3𝑥+4𝑏 𝑒^(−2𝑥) Subtracting (2) From (1) 𝑦^′′−𝑦^′=9𝑎𝑒^3𝑥+4𝑏 𝑒^(−2𝑥)−3𝑎𝑒^3𝑥+2𝑏 𝑒^(−2𝑥) 𝑦^′′−𝑦^′=6𝑎𝑒^3𝑥+6𝑏 𝑒^(−2𝑥) 𝑦^′′−𝑦^′=6(𝑎𝑒^3𝑥+𝑏𝑒^(−2𝑥)) 𝑦^′′−𝑦^′=6y 𝒚^′′−𝒚^′−𝟔𝒚=𝟎 is the required differential equation. (As y = 𝑎^3𝑥 + b𝑒^3𝑥)