Find the general solution of the equation dydx-y = 2x. Solution: The equation dydx-y = 2x is of the form dydx+Py = Q where P = □ and Q = □ ∴ I.F. = e∫-dx = e–x ∴ the solution of the linear d - Mathematics and Statistics

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Sum

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–xy = `-2x*"e"^-x+ 2 square + "c"`

∴ `y + square + square` = cex 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–xy = `-2x*"e"^-x+ 2 ("e"^-x)/(-1) + "c"`

∴ y + 2x + 2 = cex is the required general solution of the given differential equation

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Chapter 1.8: Differential Equation and Applications - Q.6

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