Question 2 - Formation of Differntial equation when general solution given - Chapter 9 Class 12 Differential Equations
Last updated at April 16, 2024 by Teachoo
Formation of Differntial equation when general solution given
Formation of Differntial equation when general solution given
Last updated at April 16, 2024 by Teachoo
Question 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