Question

Solve the differential equation `[e^(-2sqrtx)/sqrtx - y/sqrtx] dx/dy = 1 (x != 0).`

Answer 1

`[e^(- 2sqrtx)/sqrtx - y/sqrtx] dx/dy = 1`

or `dy/dx = e^(- 2sqrtx)/sqrtx - y/sqrtx`     ...(i)

Comparing with `dy/dx + Py = Q`

`P = 1/sqrtx, Q = e^(- 2sqrtx)/sqrtx`

∵ `I.F. = e^(x^(-1/2)) = e^(int 1/sqrtx dx) = e^(2sqrtx)`

Hence, the general solution of the equation,

`y * e^(2sqrtx) = int (e^(- 2sqrtx))/sqrtx * e^(2sqrtx) dx + C`

`y * e^(2sqrtx) = int 1/sqrtx dx + C`

`=> ye^(2sqrtx) = 2sqrtx + C`