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प्रश्न
Find the differential equation by eliminating arbitrary constants from the relation y = (c1 + c2x)ex
उत्तर
y = (c1 + c2x)ex
∴ ye–x = c1 + c2x
Differentiating w.r.t. x, we get
`y(-"e"^-x) + "e"^-x ("d"y)/("d"x)` = 0 + c2
∴ `"e"^x (("d"y)/("d"x) - y)` = c2
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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