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Question
Solve `("d"y)/("d"x) + y cos x + x = 2 cos x`
Solution
`("d"y)/("d"x) + y cos x + x = 2 cos x`
This is of the form `("d"y)/("d"x) + "P"y` = Q
Here P = cos x and Q = 2 cos x
`int "Pd"x = int cos x "dx` = sin x
I.F = `"e"^(intpdx)`
= `"e"^(sinx)`
The solution is
y(I.F) = `int "Q" ("I.F") "d"x+ "c"`
yesin x = `int(2 cos x) "e"^(sin x) "d"x`
yesin x = `2int"e"^"t" "dt"`
= 2`"e"^"t" + "c"`
yesin x = `2"e"^(sinx) + "c"`
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