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Question
Solve the following differential equation:
`"dx"/"dy" + "8x" = 5"e"^(- 3"y")`
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Solution
`"dx"/"dy" + "8x" = 5"e"^(- 3"y")`
∴ `"dx"/"dy" + "2x" = 5/4"e"^(- 3"y")` .....(1)
This is the linear differential equation of the form
`"dx"/"dy" + "Px" = "Q"` where P = 2 and `"Q" = 5/4 "e"^(- 3"y")`
∴ I.F. = `"e"^(int "P dy") = "e"^(2 "dy") = "e"^("2y")`
∴ the solution of (1) is given by
`"x" * ("I.F.") = int "Q" * ("I.F.") "dy" + "c"_1`
∴ `"x" * "e"^(2"y") = int 5/4 "e"^(- 3"y") * "e"^"2y" "dy" + "c"_1`
∴ `"x" * "e"^(2"y") = 5/4 int "e"^-"y" "dy" + "c"_1`
∴ `"x" "e"^(2"y") = 5/4 * ("e"^-"y")/-1 + "c"_1`
∴ 4xe2y = - 5e-y + 4c1
∴ 4xe2y + - 5e-y = c, where c = 4c1
This is the general solution.
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