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Question
For the given below, verify that the given function (implicit or explicit) is a solution to the corresponding differential equation.
`y = xsin 3x : (d^2y)/(dx^2) + 9y - 6 cos 3x = 0`
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Solution
Given function y = x sin 3x .....(1)
On differentiating with respect to x, .....(2)
`dy/dx = 3x cos 3x + 1 * sin 2x`
On differentiating again,
`(d^2y)/dx^2 = (3 cos 3x - 9x sin 3x) + 3 cos 3x`
= 6 cos 3x - 9x sin 3x ....(3)
= 6 cos 3x - 9x
`=> (d^2y)/dx^2 + 9y - 6 cos 3x = 0`
= (6 cos 3x - 9x sin 3x) + 9 (x sin 3x) - 6 cos 3x = 0 ...[Using (1) & (3)]
Hence, (1) is a solution of `(d^2y)/(dx^2) + 9y - 6 cos 3x = 0`
Hence, the given function is a solution to the differential equation.
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