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Question
Solution of the differential equation `("d"y)/("d"x) + y/x` = sec x is ______.
Options
x(y + cosx) = sinx + c
x(y – cosx) = sinx + c
xy cosx = sinx + c
x(y + cosx) = cosx + c
Solution
Solution of the differential equation `("d"y)/("d"x) + y/x` = sec x is x(y + cosx) = sinx + c.
Explanation:
The given differential equation is `("d"y)/("d"x) + y/x` = sec x
Since, it is a linear differential equation
∴ P = `1/x` and Q = sin x
Integrating factor I.F. = `"e"^(int 1/x "d"x)`
= `"e"^(log x)`
= x
∴ Solution is `y xx "I"."F" = int "Q" xx "I"."F". "d"x + "c"`
`y xx x = int sinx . x "d"x + "c"`
⇒ `y xx x = int x sin x "d"x + "c"`
⇒ `yx = x . int sinx "d"x - int("D"(x)intsinx "d"x)"d"x + "c"`
⇒ `yx = x(- cos x) - int - cos x "d"x`
⇒ `yx = - x cosx + int cosx "d"x`
⇒ `yx = -x cosx + sinx + "c"`
⇒ `yx + cosx = sinx + "c"`
⇒ `x(y + cosx) = sinx + "c"`
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