Let F(x) be the particular integral of the differential equation y" + y = (x - cot x) If there exist c e R such that F(c) = c, then c is equal to,a)π/2b)0c)π/4d)None of these.Correct answer is option 'A'. Can you explain this answer?

Question

Let F(x) be the particular integral of the differential equation y" + y = (x - cot x) If there exist c e R such that F(c) = c, then c is equal to,a)&pi;/2b)0c)&pi;/4d)None of these.Correct answer is option 'A'. Can you explain this answer?

Answer

Then by variation of parameter method, particular integral is given by

P.l. = cos x f(x) + sin x g(x) ....(1)

Here

= x sin x + cos x - In ( cosecx - cotx) - cos x

= x sin x - In (cosecx - cotx)

So by (1), then P.l. is given by,

F(x) = cos x [ x cos x] + sin x [ x sin x - In ( cosecx - cotx) ]

F(x) = x - sin x In ( cosecx - cotx)

F(0) = Not exirt.

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