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Question 10 - CBSE Class 12 Sample Paper for 2018 Boards - Solutions of Sample Papers and Past Year Papers - for Class 12 Boards

Last updated at March 16, 2023 by

Verify that ax 2 + by 2 = 1 is a solution of the differential equation x(yy 2 + y 1 2 ) = yy 1

This is a question of CBSE Sample Paper - Class 12 - 2017/18.

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Question 10 - CBSE Class 12 Sample Paper for 2018 Boards - Part 2
Question 10 - CBSE Class 12 Sample Paper for 2018 Boards - Part 3

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Transcript

Question 10 Verify that ax2 + by2 = 1 is a solution of the differential equation x(yy2 + y12) = yy1 Given ax2 + by2 = 1 First we find y1, and y2 Now, ax2 + by2 = 1 Differentiating w.r.t. x (ax2)’+ (by2)’ = (1)’ 2ax + 2by 𝑑𝑦/𝑑π‘₯ = 0 2ax + 2byy1 = 0 2(ax + byy1) = 0 ax + byy1 = 0 Now, finding y2 From (1) ax + byy1 = 0 ax + by𝑑𝑦/𝑑π‘₯ = 0 Differentiating w.r.t. x (ax)’ + ("by" 𝑑𝑦/𝑑π‘₯)^β€²= 0 a + b("y" 𝑑𝑦/𝑑π‘₯)^β€²= 0 a + b(𝑦^β€² 𝑑𝑦/𝑑π‘₯+𝑦𝑦′′)= 0 a + b(𝑦^β€² 𝑦′+𝑦𝑦′′)= 0 a + b(𝑦1 𝑦1+𝑦𝑦2)= 0 a + b(𝑦1 𝑦1+𝑦𝑦2)= 0 a + b(〖𝑦_1γ€—^2+𝑦𝑦2)= 0 a = – b(〖𝑦_1γ€—^2+𝑦𝑦2) Now, from (1) ax + byy1 = 0 Putting a = – b(〖𝑦_1γ€—^2+𝑦𝑦2) from (2) – b(〖𝑦_1γ€—^2+𝑦𝑦2)x + byy1 = 0 byy1 = b(〖𝑦_1γ€—^2+𝑦𝑦2)x Cancelling b both sides yy1 = (〖𝑦_1γ€—^2+𝑦𝑦2)x x(〖𝑦_1γ€—^2+𝑦𝑦2) = yy1 Hence proved

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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.