Question

Solve the following differential equation:

`"dy"/"dx" + "y"/"x" = "x"^3 - 3`

Answer 1

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

This is the linear differential equation of the form

`"dy"/"dx" + "P" * "y" = "Q"`, where P = `1/"x"` and Q = `"x"^3 - 3`

∴ I.F. = `"e"^(int "Pdx") = "e"^(int 1/"x" "dx")`

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

∴ the solution of (1) is given by

y(I.F.) = ∫ Q. (I.F.)dx + c₁

∴ `"y" * "x" = int ("x"^3 - 3)"x" "dx" + "c"_1`
∴ `"xy" = int ("x"^4 - 3"x") "dx" + "c"_1`

∴ `"xy" = "x"^5/5 - 3 * "x"^2/2 + "c"_1`

∴ `"x"^2/5 - "3x"^2/2 - "xy" = "c"`, where c = - c₁

∴ This is the general solution.