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प्रश्न
Solve the following Linear differential equation:
`("d"y)/("d"x) + y/x = sin x`
उत्तर
The given differential equation can be written as
`("d"y)/("d"x) + (1/x)y` = sin x
This is of the form `("d"y)/("d"x) + "P"y` = Q
Where P = `1/x`
Q = sin x
Thus, the given differential equation is linear.
I.F = `"e"^(int "pd"x)`
= `"e"^(int 1/x "d"x)`
= `"e"^logx`
= x
So, the required solution is given by
yx I.F = `int ("Q" xx "I.F") "d"x + "c"`
yx = `int sin x xx x "d"x + "c"`
= x(– cos x) – (1)(– sin x) + c
yx = – x cos x + sin x + c
yx + x cos x = sin x + c
(y + cos x)x = sin x + c is a required solution.
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