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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

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Question 2 Form a differential equation representing the given family of curves by eliminating arbitrary constants π‘Ž and 𝑏. 𝑦^2=π‘Ž(𝑏^2βˆ’π‘₯^2 ) 𝑦^2=π‘Ž(𝑏^2βˆ’π‘₯^2 ) 𝑦^2=π‘Žπ‘^2βˆ’π‘Žπ‘₯^2 Since it has two variables, we will differentiate twice ∴ Diff. Both Sides w.r.t. π‘₯ 2𝑦.𝑑𝑦/𝑑π‘₯=0βˆ’2π‘Žπ‘₯ 2𝑦𝑦^β€²=βˆ’2π‘Žπ‘₯ 𝑦𝑦′=βˆ’π‘Žπ‘₯ (𝑦𝑦^β€²)/(βˆ’π‘₯) = π‘Ž π‘Ž = (βˆ’π‘¦)/π‘₯ 𝑦′ Now, 𝑦𝑦′=βˆ’π‘Žπ‘₯ "Again Differentiating w.r.t. " π‘₯ 𝑑𝑦/𝑑π‘₯.𝑦^β€²+𝑦.(𝑑(𝑦^β€²))/𝑑π‘₯=βˆ’π‘Ž 𝑑π‘₯/𝑑π‘₯ 𝑦^′×𝑦^β€²+𝑦×𝑦^β€²β€²=βˆ’π‘Ž 〖𝑦^β€²γ€—^2+𝑦𝑦^β€²β€²=βˆ’((βˆ’π‘¦)/π‘₯ 𝑦′) 〖𝑦^β€²γ€—^2+𝑦𝑦^β€²β€²=(𝑦𝑦^β€²)/π‘₯ π‘₯〖𝑦^β€²γ€—^2+π‘₯𝑦𝑦^β€²β€²=𝑦𝑦^β€² …(1) ("Using Product Rule ") (From (1) π‘Ž= (βˆ’π‘¦)/π‘₯ 𝑑𝑦/𝑑π‘₯ ) π’™π’šπ’š^β€²β€²+π’™γ€–π’š^β€²γ€—^πŸβˆ’π’šπ’š^β€²=𝟎 which is the required differential equation

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Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.