Answer 1

`("x + a")"dy"/"dx" - 3"y" = ("x + a")^5`

∴ `"dy"/"dx" - "3y"/("x + a") = ("x + a")^4`

∴ `"dy"/"dx" + ((- 3)/("x + a"))"y" = ("x + a")^4`  ...(1)

This is the linear differential equation of the form

`"dy"/"dx" + "P"*"y" = "Q"`, where P = `(- 3)/("x + a")` and Q = (x + a)⁴ 

∴ I.F. = `"e"^(int "P dx") = "e"^(int (- 3)/("x + a")"dx") = "e"^(-3 int 1/("x + a")"dx")`

`= "e"^(- 3 log |"x + a"|) = "e"^(log ("x + a")^- 3)`

`= ("x + a")^-3 = 1/("x + a")^3`

∴ the solution of (1) is given by

`"y" * ("I.F.") = int "Q" * ("I.F.") "dx" + "c"`

∴ `"y" * 1/("x + a")^3 = int ("x + a")^4 * 1/("x + a")^3 "dx" + "c"`

∴ `"y"/("x + a")^3 = int ("x + a") "dx" + "c"`

∴ `"y"/("x + a")^3 = ("x + a"^2)/2 + "c"`

∴ 2y = (x + a)⁵ + 2c (x + a)³ 

This is the general solution.