Solve the differential equation dydx+y = e−x - Mathematics and Statistics

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Sum

Solve the differential equation `("d"y)/("d"x) + y` = e−x 

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Solution

`("d"y)/("d"x) + y` = e−x 

The given equation is of the form

`("d"y)/("d"x) + "P"y` = Q.

where P = 1 and Q = e−x 

∴ I.F. = `"e"^(int^("Pd"x))`

= `"e"^(int"d"x)`

= ex

∴ Solution of the given equation is

`y("I.""F.") = int"Q"("I.F.")  "d"x + "c"`

∴ `y * "e"^x = int"e"^(-x) xx "e"^x  "d"x + "c"`

∴ `y * "e"^x = int"e"^(-x + x)  "d"x + "c"`

∴ `y * "e"^x = int"e"^0  "d"x + "c"`

∴ `y * "e"^x = int 1"d"x + "c"`

∴ yex = x + c

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