Ex 5.7, 11 - Chapter 5 Class 12 Continuity and Differentiability

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Ex 5.7, 11 If y=5 cos⁡〖𝑥−3 sin⁡𝑥 〗 ,prove that 𝑑2𝑦/𝑑𝑥2 + y = 0 y = 5 cos⁡〖𝑥−3 sin⁡𝑥 〗 Differentiating 𝑤.𝑟.𝑡.𝑥 𝑑𝑦/𝑑𝑥 = (𝑑(5 cos⁡〖𝑥−3 sin⁡𝑥 〗))/𝑑𝑥 𝑑𝑦/𝑑𝑥 = (𝑑(5 cos⁡𝑥))/𝑑𝑥 − (𝑑(3 sin⁡𝑥))/𝑑𝑥 𝑑𝑦/𝑑𝑥 = − 5 sin⁡𝑥 − 3 cos⁡𝑥 Again Differentiating 𝑤.𝑟.𝑡.𝑥 𝑑/𝑑𝑥 (𝑑𝑦/𝑑𝑥) = (𝑑 〖(− 5 sin〗⁡𝑥 〖− 3cos〗⁡〖𝑥)〗)/𝑑𝑥 (𝑑^2 𝑦)/(𝑑𝑥^2 ) = −(𝑑(5 sin⁡𝑥))/𝑑𝑥 − (𝑑(3 cos⁡𝑥))/𝑑𝑥 (𝑑^2 𝑦)/(𝑑𝑥^2 ) = − 5 cos⁡𝑥 − 3 〖(−sin〗⁡〖𝑥)〗 (𝑑^2 𝑦)/(𝑑𝑥^2 ) = − 5 cos⁡𝑥 + 3 sin⁡𝑥 (𝑑^2 𝑦)/(𝑑𝑥^2 ) = − (5 cos⁡𝑥 − 3 sin⁡𝑥) (𝑑^2 𝑦)/(𝑑𝑥^2 ) = −y (𝑑^2 𝑦)/(𝑑𝑥^2 ) + y = 0 Hence Proved As y = 5 cos⁡〖𝑥−3 sin⁡𝑥 〗