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Find the general solution of the equation `("d"y)/("d"x) - y` = 2x.

**Solution: **The equation `("d"y)/("d"x) - y` = 2x

is of the form `("d"y)/("d"x) + "P"y` = Q

where P = `square` and Q = `square`

∴ I.F. = `"e"^(int-"d"x)` = e^{–x}

∴ the solution of the linear differential equation is

ye^{–x }= `int 2x*"e"^-x "d"x + "c"`

∴ ye^{–x } = `2int x*"e"^-x "d"x + "c"`

= `2{x int"e"^-x "d"x - int square "d"x* "d"/("d"x) square"d"x} + "c"`

= `2{x ("e"^-x)/(-1) - int ("e"^-x)/(-1)*1"d"x} + "c"`

∴ ye^{–x }= `-2x*"e"^-x + 2int"e"^-x "d"x + "c"`

∴ e^{–x}y = `-2x*"e"^-x+ 2 square + "c"`

∴ `y + square + square` = ce^{x} is the required general solution of the given differential equation

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#### Solution

The equation `("d"y)/("d"x) - y` = 2x

is of the form `("d"y)/("d"x) + "P"y` = Q

where P =** – 1** and Q = **2x**

∴ I.F. = `"e"^(int-"d"x)` = e^{–x}

∴ the solution of the linear differential equation is

ye^{–x }= `int 2x*"e"^-x "d"x + "c"`

∴ ye^{–x } = `2int x*"e"^-x "d"x + "c"`

= `2{x int"e"^-x "d"x - int "e"^-x "d"x* "d"/("d"x) x "d"x} + "c"`

= `2{x ("e"^-x)/(-1) - int ("e"^-x)/(-1)*1"d"x} + "c"`

∴ ye^{–x }= `-2x*"e"^-x + 2int"e"^-x "d"x + "c"`

∴ e^{–x}y = `-2x*"e"^-x+ 2 ("e"^-x)/(-1) + "c"`

∴ y + **2x** + **2** = ce^{x} is the required general solution of the given differential equation

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