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# Solve the following differential equation: x + adydxyx + a(x + a)dydx-3y=(x + a)5 - Mathematics and Statistics

Sum

Solve the following differential equation:

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

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#### Solution

("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.

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