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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 Dec. 16, 2024 by Teachoo

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.

You can download the question paper here  https://www.teachoo.com/cbse/sample-papers/

 

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)โ€™+ (by2)โ€™ = (1)โ€™ 2ax + 2by ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ = 0 2ax + 2byy1 = 0 2(ax + byy1) = 0 ax + byy1 = 0 Now, finding y2 From (1) ax + byy1 = 0 ax + by๐‘‘๐‘ฆ/๐‘‘๐‘ฅ = 0 Differentiating w.r.t. x (ax)โ€™ + ("by" ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ)^โ€ฒ= 0 a + b("y" ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ)^โ€ฒ= 0 a + b(๐‘ฆ^โ€ฒ ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ+๐‘ฆ๐‘ฆโ€ฒโ€ฒ)= 0 a + b(๐‘ฆ^โ€ฒ ๐‘ฆโ€ฒ+๐‘ฆ๐‘ฆโ€ฒโ€ฒ)= 0 a + b(๐‘ฆ1 ๐‘ฆ1+๐‘ฆ๐‘ฆ2)= 0 a + b(๐‘ฆ1 ๐‘ฆ1+๐‘ฆ๐‘ฆ2)= 0 a + b(ใ€–๐‘ฆ_1ใ€—^2+๐‘ฆ๐‘ฆ2)= 0 a = โ€“ b(ใ€–๐‘ฆ_1ใ€—^2+๐‘ฆ๐‘ฆ2) Now, from (1) ax + byy1 = 0 Putting a = โ€“ b(ใ€–๐‘ฆ_1ใ€—^2+๐‘ฆ๐‘ฆ2) from (2) โ€“ b(ใ€–๐‘ฆ_1ใ€—^2+๐‘ฆ๐‘ฆ2)x + byy1 = 0 byy1 = b(ใ€–๐‘ฆ_1ใ€—^2+๐‘ฆ๐‘ฆ2)x Cancelling b both sides yy1 = (ใ€–๐‘ฆ_1ใ€—^2+๐‘ฆ๐‘ฆ2)x x(ใ€–๐‘ฆ_1ใ€—^2+๐‘ฆ๐‘ฆ2) = yy1 Hence proved
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  2. Solutions of Sample Papers and Past Year Papers - for Class 12 Boards

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Solutions of Sample Papers and Past Year Papers - for Class 12 Boards

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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 14 years. He provides courses for Maths, Science and Computer Science at Teachoo