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Find the differential equation by eliminating arbitrary constants from the relation y = (c_{1} + c_{2}x)e^{x}

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#### Solution

y = (c_{1} + c_{2}x)e^{x}

∴ ye^{–x} = c_{1} + c_{2}x

Differentiating w.r.t. x, we get

`y(-"e"^-x) + "e"^-x ("d"y)/("d"x)` = 0 + c_{2}

∴ `"e"^x (("d"y)/("d"x) - y)` = c_{2}

Again, differentiating w.r.t. x, we get

`"e"^-x (("d"^2y)/("d"x^2) - ("d"y)/("d"x)) - "e"^-x (("d"y)/("d"x) - y)` = 0

∴ `"e"^-x (("d"^2y)/("d"x^2) - ("d"y)/("d"x) - ("d"y)/("d"x) + y)` = 0

∴ `("d"^2y)/("d"x^2) - 2("d"y)/("d"x) + y` = 0

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