Question

Solve:

`xsinx dy/dx + (xcosx + sinx)y` = sin x

Answer 1

Given equation can be written as

`dy/dx + ((xcosx + sinx)y)/(xsinx) = sinx/(xsinx)`

∴ `dy/dx + (cot x + 1/x)y = 1/x`

This is a linear differential equation of the form

`dy/dx + Py` = Q

∴ P = `cot x + 1/x`,

Q = `1/x`

∴ I.F. = `e^(intPdx)`

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

= `e^(logsinx + logx)`

= `e^(logxsinx)`

= x sin x

∴ The solution is given by

y(I.F.) = `intQ*(I.F.)dx + c`

`y*xsinx = int1/x*xsinx  dx + c`

= `intsinx  dx + c`

= `- cosx + c`

∴ xy sin x + cos x = c