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Ex 9.3, 2 - Form differential equation: y2 = a (b2 - x2)

Ex 9.3, 2 - Chapter 9 Class 12 Differential Equations - Part 2
Ex 9.3, 2 - Chapter 9 Class 12 Differential Equations - Part 3


Transcript

Ex 9.3, 2 Form a differential equation representing the given family of curves by eliminating arbitrary constants 𝑎 and 𝑏. 𝑦^2=𝑎(𝑏^2−𝑥^2 ) 𝑦^2=𝑎(𝑏^2−𝑥^2 ) 𝑦^2=𝑎𝑏^2−𝑎𝑥^2 Since it has two variables, we will differentiate twice ∴ Diff. Both Sides w.r.t. 𝑥 2𝑦.𝑑𝑦/𝑑𝑥=0−2𝑎𝑥 2𝑦𝑦^′=−2𝑎𝑥 𝑦𝑦′=−𝑎𝑥 (𝑦𝑦^′)/(−𝑥) = 𝑎 𝑎 = (−𝑦)/𝑥 𝑦′ Now, 𝑦𝑦′=−𝑎𝑥 "Again Differentiating w.r.t. " 𝑥 𝑑𝑦/𝑑𝑥.𝑦^′+𝑦.(𝑑(𝑦^′))/𝑑𝑥=−𝑎 𝑑𝑥/𝑑𝑥 𝑦^′×𝑦^′+𝑦×𝑦^′′=−𝑎 〖𝑦^′〗^2+𝑦𝑦^′′=−((−𝑦)/𝑥 𝑦′) 〖𝑦^′〗^2+𝑦𝑦^′′=(𝑦𝑦^′)/𝑥 𝑥〖𝑦^′〗^2+𝑥𝑦𝑦^′′=𝑦𝑦^′ …(1) ("Using Product Rule ") (From (1) 𝑎= (−𝑦)/𝑥 𝑑𝑦/𝑑𝑥 ) 𝒙𝒚𝒚^′′+𝒙〖𝒚^′〗^𝟐−𝒚𝒚^′=𝟎 which is the required differential equation

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Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.