Question

Find the general solution of `"dy"/"dx" + "a"y` = e^mx 

Answer 1

Given equation is `"dy"/"dx" + "a"y` = e^mx 

Here, P = a and Q = e^mx 

∴ I.F. = `"e"^(int Pdx)`

= `"e"^(int a .dx)`

= e^ax.

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)`