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Question
If y = 5 cos x – 3 sin x, prove that `(d^2y)/(dx^2) + y = 0`
Solution
Given, y = 5 cos x - 3 sin x
Differentiating both sides with respect to x,
`dy/dx = 5 d/dx cos x - 3 d/dx sin x`
= 5 (- sin x) - 3 cos x = - 5 sin x - 3 cos x
Differentiating both sides again with respect to x,
`(d^2 y)/dx= - 5 d/dx sin x - 3 d/dx cos x`
= - 5 cos x - 3 (- sin x) = 3 sin x - 5 cos x
Hence, `(d^2 y)/dx^2 + y ` = 3 sin x - 5 cos x + 5 cos x - 3 sin x ...(On substituting the value of y)
= 0
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