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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


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 is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths and Science at Teachoo.