Question

Solve the differential equation dy = cosx(2 – y cosecx) dx given that y = 2 when x = `pi/2`

Answer 1

The given differential equation is dy = cosx(2 – y cosecx) dx

⇒ `"dy"/"dx"` = cosx(2 – y cosec x)

⇒ `"dy"/"dx"` = 2cosx – ycosx . cosecx

⇒ `"dy"/"dx"` = 2cosx – ycotx

⇒ `"dy"/"dx" + y cot x` = 2cosx 

Here, P = cotx and Q = 2cosx.

∴ Integrating factor I.F. = `"e"^(intPdx)`

= `"e"^(int cot xdx)`

= `"e"^(log sinx)`

= sin x

∴ Required solution is `y xx "I"."F" = int "Q" xx "I"."F".  "d"x + "c"`

⇒ `y . sin x = int 2 cos x . sin x "d"x + "c"`

⇒ `y . sin x = int sin 2x  "d"x + "c"`

⇒ `y . sin x = - 1/2 cos 2x + "c"`

Put x = `pi/2` and y = 2, we get

`2 sin  pi/2 = - 1/2 cos  pi + "c"`

⇒  2(1) = `- 1/2 (-1) + "c"`

⇒  2 = `1/2 + "c"`

⇒ c = `2 - 1/2 = 3/2`

∴ The equation is y sin x = `- 1/2 cos 2x + 3/2`.