# Example 41

Last updated at March 11, 2017 by Teachoo

Last updated at March 11, 2017 by Teachoo

Transcript

Example 41 If y = 𝑠𝑖𝑛−1𝑥, show that (1 – 𝑥2) 𝑑2𝑦𝑑𝑥2 − 𝑥 𝑑𝑦𝑑𝑥 = 0 . We have 𝑦 = 𝑠𝑖𝑛−1𝑥 Differentiating 𝑤.𝑟.𝑡.𝑥 𝑑𝑦𝑑𝑥 = 𝑑 𝑠𝑖𝑛−1𝑥𝑑𝑥 𝑑𝑦𝑑𝑥 = 1 1 − 𝑥2 𝒅𝒚𝒅𝒙 = 𝟏−𝒙𝟐 −𝟏 𝟐 Now, 𝑑𝑦𝑑𝑥 = 1−𝑥2 −1 2 Again Differentiating 𝑤.𝑟.𝑡.𝑥 𝑑𝑑𝑥 𝑑𝑦𝑑𝑥 = 𝑑 1 − 𝑥2 −1 2𝑑𝑥 𝑑2𝑦 𝑑𝑥2 = −1 2 1−𝑥2 −1 2 −1 . 𝑑 1 − 𝑥2𝑑𝑥 𝑑2𝑦 𝑑𝑥2 = −1 2 1−𝑥2 −32 . 𝑑(1)𝑑𝑥 − 𝑑( 𝑥2)𝑑𝑥 𝑑2𝑦 𝑑𝑥2 = −1 2 1−𝑥2 −32 . 0−2𝑥 𝑑2𝑦 𝑑𝑥2 = −1 2 1−𝑥2 −32 . −2𝑥 𝒅𝟐𝒚 𝒅𝒙𝟐 = −𝒙 𝟏−𝒙𝟐 −𝟑𝟐 Now, we need to prove 1−𝑥2 𝑑2𝑦 𝑑𝑥2 − 𝑥 . 𝑑𝑦𝑑𝑥 = 0 Solving LHS 1−𝑥2 𝑑2𝑦 𝑑𝑥2 − 𝑥 . 𝑑𝑦𝑑𝑥 = 1−𝑥2 . 𝑥 1−𝑥2 −32 − 𝑥 1−𝑥2 −1 2 = 𝑥 1−𝑥21+ −32 −𝑥 1−𝑥2 −1 2 = 𝑥 1−𝑥2 −1 2−𝑥 1−𝑥2 −1 2 = 0 = RHS Hence proved .

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About the Author

Davneet Singh

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He provides courses for Mathematics from Class 9 to 12. You can ask questions here.