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Question
Find the general solution of `"dy"/"dx" + "a"y` = emx
Solution
Given equation is `"dy"/"dx" + "a"y` = emx
Here, P = a and Q = emx
∴ I.F. = `"e"^(int Pdx)`
= `"e"^(int a .dx)`
= eax.
Solution of equation is `y xx "I"."F" = int "Q" "I"."F" "d"x + "c"`
⇒ `y."e"^("a"x) = int "e"^"mx" . "e"^("a"x) "d"x + "c"`
⇒ `y . "e"^("a"x) = int "e"^(("m" + "a")x) "d"x + "c"`
⇒ `y . "e"^("a"x) = "e"^(("m" + "a")x)/(("m" + "a")) + "c"`
⇒ y = `"e"^(("m" + "a")x)/(("m" + "a")) . "e"^(-"a"x) + "c"."e"^(-"a"x)`
∴ y = `"e"^("m"x)/(("m" + "a")) + "c" . "e"^(-"a"x)`
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