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

Answer 1

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"`