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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 = e^x (acos x + b sin x) : (d^2y)/(dx^2) - 2 dy/dx + 2y = 0`
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Solution
Given function y = ex (a cos x + b sin x)
On differentiating
`dy/dx = e^x (a sin x + b cos x) + e^x (a cos x + b sin x)`
`= e^x [(a + b) cos x + (b - a) sin x]`
On differentiating again,
`(d^2y)/dx^2 = e^x [- (a + b) sin x + (b - a) cos x] + e^x [(a + b) cos x + (b - a) sin x]`
`(d^2y)/(dx^2)= e^x [2b cos x - 2a sin x]`
L.H.S. ⇒ `(d^2y)/(dx^2) - 2 dy/dx + 2y`
`= e^x [2b cos x - 2a sin x] - 2e^x [(a + b) cos x + (b - a) sin x] + 2e^x [a cos x + b sin x] = 0`
`= e^x [(2b - 2a - 2b + 2a)] cos x + (- 2a - 2b + 2a + 2b) sin x`
`= "e"^x xx 0 + 0 xx sin x = 0` R.H.S.
Hence the given function is a solution to the differential equation.
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