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Ex 5.7, 13 - If y = 3 cos (log x) + 4 sin (log x), show - Ex 5.7

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  1. Chapter 5 Class 12 Continuity and Differentiability
  2. Serial order wise
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Ex 5.7, 13 If 𝑦=3 cos﷮ ( log﷮𝑥)+4 sin﷮ ( log﷮𝑥 )﷯﷯﷯﷯, show that 𝑥2 𝑦2 + 𝑥𝑦1 + 𝑦 = 0 𝑦=3 cos﷮ ( log﷮𝑥)+4 sin﷮ ( log﷮𝑥)﷯﷯﷯﷯ Differentiating 𝑤.𝑟.𝑡.𝑥 𝑑𝑦﷮𝑑𝑥﷯ = 𝑑(3 cos﷮ ( log﷮𝑥) + 4 sin﷮ ( log﷮𝑥)﷯﷯﷯﷯)﷮𝑑𝑥﷯ 𝑑𝑦﷮𝑑𝑥﷯ = 𝑑 3 cos﷮( log﷮𝑥﷯﷯﷯)﷮𝑑𝑥﷯ + 𝑑 4 𝑠𝑖𝑛 ﷮( log﷮𝑥﷯﷯﷯)﷮𝑑𝑥﷯ = 3 𝑑 cos﷮( log﷮𝑥﷯﷯﷯)﷮𝑑𝑥﷯ + 4 𝑑 𝑠𝑖𝑛 ﷮( log﷮𝑥﷯﷯﷯)﷮𝑑𝑥﷯ = 3 − sin﷮ log﷮𝑥﷯﷯ . 𝑑 ( log﷮𝑥) ﷯﷮𝑑𝑥﷯﷯﷯ + 4 cos﷮( log﷮𝑥) . 𝑑 ( log﷮𝑥)﷯﷮𝑑𝑥﷯﷯﷯﷯ = 3 − sin﷮ log﷮𝑥﷯﷯ . 1﷮𝑥﷯﷯﷯ + 4 cos﷮( log﷮𝑥) . 1﷮𝑥﷯﷯﷯﷯ = − 3 sin﷮( log﷮𝑥)﷯﷯﷮𝑥﷯ + 4 cos﷮( log﷮𝑥) ﷯﷯﷮𝑥﷯ Thus , 𝑦1 = 𝑑𝑦﷮𝑑𝑥﷯ = −3 sin﷮( log﷮𝑥)﷯﷯﷮𝑥﷯ + 4 cos﷮( log﷮𝑥) ﷯﷯﷮𝑥﷯ Again Differentiating 𝑤.𝑟.𝑡.𝑥 𝑑﷮𝑑𝑥﷯ 𝑑𝑦﷮𝑑𝑥﷯﷯ = 𝑦2 = 𝑑﷮𝑑𝑥﷯ −3 sin﷮( log﷮𝑥)﷯﷯﷮𝑥﷯ + 4 cos﷮( log﷮𝑥) ﷯﷯﷮𝑥﷯﷯ 𝑦2 = −3 𝑑﷮𝑑𝑥﷯ sin﷮( log﷮𝑥)﷯﷯﷮𝑥﷯﷯ + 4 𝑑﷮𝑑𝑥﷯ cos﷮( log﷮𝑥) ﷯﷯﷮𝑥﷯﷯ 𝑦2 = −3 𝑑 sin﷮( log﷮𝑥)﷯﷯﷯ ﷮𝑑𝑥﷯ . 𝑥 − 𝑑 𝑥 ﷯﷮𝑑𝑥﷯ . sin﷮( log﷮𝑥)﷯﷯﷮ ﷮1 − 𝑥﷮2﷯ ﷯ ﷯﷮2﷯﷯ ﷯ + 4 𝑑 cos ﷮( log﷮𝑥)﷯﷯﷯ ﷮𝑑𝑥﷯ . 𝑥 − 𝑑 𝑥 ﷯﷮𝑑𝑥﷯ . cos ﷮( log﷮𝑥)﷯﷯﷮ ﷮1 − 𝑥﷮2﷯ ﷯ ﷯﷮2﷯﷯ ﷯ 𝑦2 = −3 cos﷮ ( log﷮𝑥) ﷯﷯. 𝑑 log﷮𝑥﷯﷯ ﷮𝑑𝑥﷯ . 𝑥 − 1 . sin﷮( log﷮𝑥)﷯﷯﷮ 𝑥﷮2﷯﷯ ﷯ + 4 −sin﷮ ( log﷮𝑥) ﷯﷯. 𝑑 log﷮𝑥﷯﷯ ﷮𝑑𝑥﷯ . 𝑥 − 1 . cos ﷮( log﷮𝑥)﷯﷯﷮ 𝑥﷮2﷯﷯ ﷯ 𝑦2 = −3 cos﷮ ( log﷮𝑥) ﷯﷯. 1﷮𝑥﷯ . 𝑥 − sin﷮( log﷮𝑥)﷯﷯﷮ 𝑥﷮2﷯﷯ ﷯ + 4 −sin﷮ ( log﷮𝑥) ﷯﷯. 1﷮𝑥﷯ . 𝑥 − cos ﷮( log﷮𝑥)﷯﷯﷮ 𝑥﷮2﷯﷯ ﷯ 𝑦2 = −3 cos﷮ ( log﷮𝑥) ﷯﷯− sin﷮( log﷮𝑥)﷯﷯﷮ 𝑥﷮2﷯﷯ ﷯ + 4 −sin﷮ ( log﷮𝑥) ﷯﷯− cos ﷮( log﷮𝑥)﷯﷯﷮ 𝑥﷮2﷯﷯ ﷯ 𝑦2 = −3 cos﷮ ( log﷮𝑥) ﷯﷯+ 3 sin﷮( log﷮𝑥)﷯﷯ − 4 sin﷮( log﷮𝑥) ﷯− 4 cos﷮( log﷮𝑥 )﷯﷯ ﷯﷮ 𝑥﷮2﷯﷯ 𝑦2 = −7 cos﷮ log﷮𝑥﷯﷯﷯ − sin﷮( log﷮𝑥﷯﷯)﷮ 𝑥﷮2﷯﷯ Hence 𝒅﷮𝟐﷯𝒚﷮𝒅 𝒙﷮𝟐﷯﷯ = 𝒚𝟐 = −𝟕 𝒄𝒐𝒔﷮ 𝒍𝒐𝒈﷮𝒙﷯﷯﷯ − 𝒔𝒊𝒏﷮( 𝒍𝒐𝒈﷮𝒙﷯﷯)﷮ 𝒙﷮𝟐﷯﷯ We need to prove 𝑥2 𝑦2 + 𝑥𝑦1 + 𝑦 = 0 Solving L.H.S 𝑥2 −7 cos﷮ log﷮𝑥﷯﷯﷯ − sin﷮( log﷮𝑥﷯﷯)﷮ 𝑥﷮2﷯﷯﷯ + 𝑥 −3 sin﷮ log﷮𝑥﷯﷯﷯ +4 cos﷮( log﷮𝑥﷯﷯)﷮𝑥﷯﷯ + 3 cos﷮ log﷮𝑥﷯﷯﷯ +4 sin﷮ log﷮𝑥﷯﷯ ﷯ = −7 cos﷮ log﷮𝑥﷯﷯﷯ − sin﷮( log﷮𝑥﷯﷯)− 3 sin﷮ log﷮𝑥﷯﷯﷯ +4 cos﷮( log﷮𝑥﷯﷯) + 3 cos﷮ log﷮𝑥﷯﷯﷯ +4 sin﷮ log﷮𝑥﷯﷯ ﷯ = −7 cos﷮ log﷮𝑥﷯﷯﷯+7 cos﷮ log﷮𝑥﷯﷯﷯−4 sin﷮( log﷮𝑥﷯﷯)+4 cos﷮( log﷮𝑥﷯﷯) = 0 − 0 = 0 = RHS Hence proved

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