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Question
Find the differential equation by eliminating arbitrary constants from the relation y = (c1 + c2x)ex
Solution
y = (c1 + c2x)ex ......(i)
Here, c1 and c2 are arbitrary constants.
Differentiating w.r.t. x, we get
`("d"y)/("d"x)` = (c1 + c2x)ex + c2ex
∴ `("d"y)/("d"x)` = y + c2ex ......(ii) .......[From(i)]
Again, differentiating w.r.t. x, we get
`("d"^2y)/("d"x^2) = ("d"y)/("d"x) + "c"_2"e"^x`
∴ c2ex = `("d"^2y)/("d"x^2) - ("d"y)/("d"x)` .....(iii)
Substituting (iii) in (ii), we get
`("d"y)/("d"x) = y + ("d"^2y)/("d"x^2) - ("d"y)/("d"x)`
∴ `("d"^2y)/("d"x^2) - 2 ("d"y)/("d"x) + y` = 0
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