# Misc 17

Last updated at Dec. 8, 2016 by Teachoo

Last updated at Dec. 8, 2016 by Teachoo

Transcript

Misc 17 If 𝑥=𝑎 (cos𝑡 + 𝑡 sin𝑡) and y=𝑎 (sin𝑡 – 𝑡 cos𝑡), Find 𝑑2𝑦 𝑑𝑥2 If 𝑥=𝑎 (cos𝑡 + 𝑡 sin𝑡) & 𝑦=𝑎 (sin𝑡 – 𝑡 cos𝑡) We need to find 𝑑2𝑦 𝑑𝑥2 First we find 𝑑𝑦𝑑𝑥 𝑑𝑦𝑑𝑥 = 𝑑𝑦𝑑𝑥 . 𝑑𝑡𝑑𝑡 𝑑𝑦𝑑𝑥 = 𝑑𝑦𝑑𝑡 . 𝑑𝑡𝑑𝑥 𝑑𝑦𝑑𝑥 = 𝑑𝑦𝑑𝑡 𝑑𝑥𝑑𝑡 Calculating 𝒅𝒚/𝒅𝒕 𝑦=𝑎 sin𝑡– 𝑡 cos𝑡 Differentiating 𝑤.𝑟.𝑡.𝑥. 𝑑𝑦𝑑𝑡 = 𝑑 𝑎 sin𝑡– 𝑡 cos𝑡𝑑𝑡 𝑑𝑦𝑑𝑡 = 𝑎 𝑑 sin𝑡– 𝑡 cos𝑡𝑑𝑡 𝑑𝑦𝑑𝑡 = 𝑎 𝑑 sin𝑡𝑑𝑡 − 𝑑 𝑡 cos𝑡𝑑𝑡 𝑑𝑦𝑑𝑡 = 𝑎 cos𝑡 − 𝑑 𝑡 cos𝑡𝑑𝑡 𝑑𝑦𝑑𝑡 = 𝑎 cos𝑡 − 𝑑𝑡𝑑𝑡 . cos𝑡+ 𝑑 cos𝑡𝑑𝑡 . 𝑡 𝑑𝑦𝑑𝑡 = 𝑎 cos𝑡 − cos𝑡+ −sin𝑡 . 𝑡 𝑑𝑦𝑑𝑡 = 𝑎 cos𝑡 − cos𝑡− sin𝑡 . 𝑡 𝑑𝑦𝑑𝑡 = 𝑎 cos𝑡 − cos𝑡+𝑡 . sin𝑡 𝑑𝑦𝑑𝑡 = 𝑎 0+𝑡 sin𝑡 𝑑𝑦𝑑𝑡 = 𝑎 .𝑡. sin𝑡 Hence 𝑑𝑦/𝑑𝑡 = 𝑎 .𝑡. sin𝑡 Calculating 𝒅𝒙/𝒅𝒕 𝑥=𝑎 (cos𝑡 + 𝑡 sin𝑡) Differentiating 𝑤.𝑟.𝑡.𝑥. 𝑑𝑥𝑑𝑡 = 𝑑 𝑎 (cos𝑡 + 𝑡 sin𝑡) 𝑑𝑡 𝑑𝑥𝑑𝑡 = 𝑎 𝑑 cos𝑡 + 𝑡 sin𝑡𝑑𝑡 𝑑𝑥𝑑𝑡 = 𝑎 𝑑 cos𝑡𝑑𝑡 + 𝑑 𝑡 sin𝑡𝑑𝑡 𝑑𝑥𝑑𝑡 = 𝑎 −sin𝑡 + 𝑑 𝑡 sin𝑡𝑑𝑡 𝑑𝑥𝑑𝑡 = 𝑎 −sin𝑡+ 𝑑𝑡𝑑𝑡 . sin𝑡+ 𝑑 sin𝑡𝑑𝑡 . 𝑡 𝑑𝑥𝑑𝑡 = 𝑎 −sin𝑡+ sin𝑡+ cos𝑡 . 𝑡 𝑑𝑥𝑑𝑡= 𝑎 − sin𝑡+ sin𝑡+𝑡 . c𝑜𝑠𝑡 𝑑𝑥𝑑𝑡 = 𝑎 .𝑡. cos𝑡 Now , 𝑑𝑦𝑑𝑥 = 𝑑𝑦/𝑑𝑡𝑑𝑥/𝑑𝑡 𝑑𝑦𝑑𝑥 = 𝑎 .𝑡. sin𝑡𝑎 .𝑡. cos𝑡 𝒅𝒚𝒅𝒙 = 𝒕𝒂𝒏𝒕 Again Differentiating 𝑤.𝑟.𝑡.𝑥. 𝑑𝑑𝑥 𝑑𝑦𝑑𝑥 = 𝑑 tan𝑡𝑑𝑥 𝑑2𝑦𝑑 𝑥2 = 𝑑 tan𝑡𝑑𝑥 𝑑2𝑦𝑑 𝑥2 = 𝑑 tan𝑡𝑑𝑥 . 𝑑𝑡𝑑𝑡 𝑑2𝑦𝑑 𝑥2 = sec2𝑡 . 𝑑𝑡𝑑𝑥 𝑑2𝑦𝑑 𝑥2 = sec2𝑡 ÷ 𝒅𝒙𝒅𝒕 𝑑2𝑦𝑑 𝑥2 = sec2𝑡 ÷ 𝒂.𝒕. 𝐜𝐨𝐬𝒕 𝑑2𝑦𝑑 𝑥2 = sec2𝑡 𝑎 . 𝑡. cos𝑡 𝑑2𝑦𝑑 𝑥2 = sec2𝑡 𝑎 . 𝑡 × 1 sec𝑡 𝑑2𝑦𝑑 𝑥2 = sec3𝑡 𝑎 . 𝑡 Hence 𝒅𝟐𝒚𝒅 𝒙𝟐 = 𝒔𝒆𝒄𝟑𝒕 𝒂 . 𝒕

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Davneet Singh

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