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Ex 9.2, 6 - Verify y = x sin x, xy' = y + x root x2 - y2

Ex 9.2, 6 - Chapter 9 Class 12 Differential Equations - Part 2

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Ex 9.2, 6 Verify that the given functions (explicit or implicit) is a solution of the corresponding differential equation : 𝑦=π‘₯ sin⁑π‘₯: π‘₯𝑦^β€²=𝑦+π‘₯√(π‘₯^2βˆ’π‘¦^2 ) (π‘₯β‰ 0 π‘Žπ‘›π‘‘ π‘₯>𝑦 π‘œπ‘Ÿ π‘₯<βˆ’π‘¦ ) 𝑦=π‘₯ sin⁑π‘₯ 𝑑𝑦/𝑑π‘₯=(𝑑(π‘₯ sin⁑π‘₯))/𝑑π‘₯ =𝑑(π‘₯)/𝑑π‘₯.sin⁑π‘₯+π‘₯.𝑑(sin⁑π‘₯ )/𝑑π‘₯ =1.sin⁑π‘₯+π‘₯(cos⁑π‘₯ ) =sin⁑π‘₯+π‘₯cos⁑π‘₯ Now, we have to verify π‘₯𝑦^β€²=𝑦+π‘₯√(π‘₯^2βˆ’π‘¦^2 ) L.H.S π‘₯𝑦^β€² =π‘₯[sin π‘₯+π‘₯π‘π‘œπ‘  π‘₯] =π‘₯ 𝑠𝑖𝑛 π‘₯+π‘₯^2 cos π‘₯ R.H.S 𝑦+π‘₯√(π‘₯^2βˆ’π‘¦^2 ) =π‘₯ 𝑠𝑖𝑛 π‘₯+π‘₯√(π‘₯^2βˆ’(π‘₯ 𝑠𝑖𝑛 π‘₯)^2 ) =π‘₯ 𝑠𝑖𝑛 π‘₯+π‘₯√(π‘₯^2 (1βˆ’sin^2⁑π‘₯ ) ) =π‘₯ 𝑠𝑖𝑛 π‘₯+π‘₯^2 √(cos^2⁑π‘₯ ) =π‘₯ 𝑠𝑖𝑛 π‘₯+π‘₯^2 cos⁑π‘₯ Since LHS = R.H.S Hence Verified

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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.