Example 36 - Chapter 5 Class 12 Continuity and Differentiability

Transcript

Example 36 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 (𝑑^2 𝑦)/〖𝑑𝑥〗^2 = – (A sin⁡𝑥 + B cos⁡𝑥) (𝑑^2 𝑦)/〖𝑑𝑥〗^2 = –y 𝒅𝟐𝒚/𝒅𝒙𝟐 + 𝒚 = 𝟎 Hence proved (As 𝑦 = 𝐴 𝑠𝑖𝑛⁡𝑥+𝐵 𝑐𝑜𝑠⁡𝑥)