Question

Solve the following Linear differential equation:

`x sin x ("d"y)/("d"x) + (x cos x + sin x)y = sinx`

Answer 1

The given differential equation can be written as

`(xsinx)/(xsinx) ("d"y)/("d"x) + ((xcosx + sinx)y)/(xsinx) = sinx/(xsinx)`

`("d"y)/("d"x) + ((xcosx)/(xsinx) + sinx/(xsinx))y = 1/x`

`("d"y)/("d"x) + (cot x + 1/x)y = 1/x`

This is of the form `("d"y)/("d"x) + "P"y` = Q

Where P = `cot x + 1/x`

Q = `1/x`

Thus, the given differential equation is linear.

I.F = `"e"^(int "pd"x)`

= `"e"^(int (cotx + 1/x)"d"x)`

= `"e"^(log sinx + logx)`

= `"e"^(log(xsinx)`

= x sin x

So, the solution of the given differential equation is given by

y × I.F = `int ("Q" xx "I.F")  "d"x + "c"`

y(x sin x) = `int 1/x x sin x  "d"x + "c"`

= `int sin x  "d"x + "c"`

y(x sin x) = − cos x + c

xy sin x + cos x = c is the required solution.