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Example 39 - If y = A sin x + B cos x, prove d2y/dx2 + y = 0

Example 39 - Chapter 5 Class 12 Continuity and Differentiability - Part 2

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Example 39 If 𝑦 = A sin⁑π‘₯+B cos⁑π‘₯, then prove that 𝑑2𝑦/𝑑π‘₯2 + y = 0. 𝑦 = A sin⁑π‘₯+B cos⁑π‘₯ Differentiating 𝑀.π‘Ÿ.𝑑.π‘₯ 𝑑𝑦/𝑑π‘₯ = 𝑑(A sin⁑π‘₯ + B cos⁑π‘₯" " )/𝑑π‘₯ 𝑑𝑦/𝑑π‘₯ = 𝑑(A sin⁑π‘₯ )/𝑑π‘₯ + 𝑑(B cos⁑π‘₯ )/𝑑π‘₯ 𝑑𝑦/𝑑π‘₯ = A . 𝑑(sin⁑π‘₯ )/𝑑π‘₯ + B . 𝑑(cos⁑π‘₯" " )/𝑑π‘₯ 𝑑𝑦/𝑑π‘₯ = A cos⁑π‘₯" " + B (βˆ’ sin⁑π‘₯) π’…π’š/𝒅𝒙 = A 𝒄𝒐𝒔⁑𝒙" " βˆ’ B π’”π’Šπ’β‘π’™ Again Differentiating 𝑀.π‘Ÿ.𝑑.π‘₯ (𝑑^2 𝑦)/〖𝑑π‘₯γ€—^2 = (𝑑 (γ€–A cos〗⁑π‘₯" " " βˆ’" γ€–B sin〗⁑π‘₯))/𝑑π‘₯ (𝑑^2 𝑦)/〖𝑑π‘₯γ€—^2 = 𝑑(A cos⁑π‘₯ )/𝑑π‘₯ βˆ’ 𝑑(B sin⁑π‘₯" " )/𝑑π‘₯ (𝑑^2 𝑦)/〖𝑑π‘₯γ€—^2 = –A sin⁑π‘₯ βˆ’ B cos⁑π‘₯ (𝑑^2 𝑦)/〖𝑑π‘₯γ€—^2 = – (A sin⁑π‘₯ + B cos⁑π‘₯) (𝑑^2 𝑦)/〖𝑑π‘₯γ€—^2 = –y π’…πŸπ’š/π’…π’™πŸ + π’š = 𝟎 Hence proved (As 𝑦 = 𝐴 𝑠𝑖𝑛⁑π‘₯+𝐡 π‘π‘œπ‘ β‘π‘₯)

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Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.