Example 39 - If y = A sin x + B cos x, prove d2y/dx2 + y = 0 - Finding second order derivatives- Implicit form

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  1. Chapter 5 Class 12 Continuity and Differentiability
  2. Serial order wise
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Example 39 If 𝑦 = A sin⁡𝑥+B cos⁡𝑥, then prove that 𝑑2𝑦﷮𝑑𝑥2﷯ + y = 0. 𝑦 = A sin⁡𝑥+B cos⁡𝑥 Differentiating 𝑤.𝑟.𝑡.𝑥 𝑑𝑦﷮𝑑𝑥﷯ = 𝑑 A sin﷮𝑥﷯ + B cos﷮𝑥 ﷯﷯﷮𝑑𝑥﷯ 𝑑𝑦﷮𝑑𝑥﷯ = 𝑑 A sin﷮𝑥﷯﷯﷮𝑑𝑥﷯ + 𝑑 B cos﷮𝑥﷯﷯﷮𝑑𝑥﷯ 𝑑𝑦﷮𝑑𝑥﷯ = A . 𝑑 sin﷮𝑥﷯﷯﷮𝑑𝑥﷯ + B . 𝑑 cos﷮𝑥 ﷯﷯﷮𝑑𝑥﷯ 𝑑𝑦﷮𝑑𝑥﷯ = A cos﷮𝑥 ﷯ + B (− sin﷮𝑥﷯) 𝑑𝑦﷮𝑑𝑥﷯ = A cos﷮𝑥 ﷯ − B sin﷮𝑥﷯ Again Differentiating 𝑤.𝑟.𝑡.𝑥 𝑑﷮2﷯𝑦﷮ 𝑑𝑥﷮2﷯﷯ = 𝑑 ( A cos﷮𝑥 ﷯ − B sin﷮𝑥﷯) ﷮𝑑𝑥﷯ 𝑑﷮2﷯𝑦﷮ 𝑑𝑥﷮2﷯﷯ = 𝑑 A cos﷮𝑥﷯﷯﷮𝑑𝑥﷯ + 𝑑 B sin﷮𝑥 ﷯﷯﷮𝑑𝑥﷯ 𝑑﷮2﷯𝑦﷮ 𝑑𝑥﷮2﷯﷯ = –A sin﷮𝑥﷯ + B ( cos﷮𝑥﷯) We need to show 𝑑2𝑦﷮𝑑𝑥2﷯ + 𝑦 = 0 Solving LHS 𝑑2𝑦﷮𝑑𝑥2﷯ + 𝑦 = (−A sin x − B cos x) + (A sin x + B cos x) = 0 = RHS Hence proved

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