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Question
Obtain the differential equation by eliminating the arbitrary constants from the following equation:
y = c1e2x + c2e5x
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Solution
y = c1e2x + c2e5x ....(1)
Differentiating twice w.r.t. x, we get
`"dy"/"dx" = "c"_1"e"^(2"x") xx 2 + "c"_2"e"^(5"x") xx 5`
∴ `"dy"/"dx" = 2"c"_1"e"^(2"x") + 5"c"_2"e"^(5"x")` ....(2)
and `("d"^2"y")/"dx"^2 = 2"c"_1"e"^(2"x") xx 2 + 5"c"_2"e"^(5"x") xx 5`
∴ `("d"^2"y")/"dx"^2 = 4"c"_1"e"^(2"x") + 25"c"_2"e"^("5x")` .....(3)
The equations (1), (2) and (3) are consistent in c1e2x and c2e5x
∴ determinant of their consistency is zero.
∴ `|("y",1,1),("dy"/"dx",2,5),(("d"^2"y")/"dx"^2,4,25)| = 0`
∴ y(50 - 20) - `1(25"dy"/"dx" - 5 ("d"^2"y")/"dx"^2) + 1 (4"dy"/"dx" - 2("d"^2"y")/"dx"^2) = 0`
∴ 30y - 25`"dy"/"dx" + 5("d"^2"y")/"dx"^2 + 4 "dy"/"dx" - 2("d"^2"y")/"dx"^2 = 0`
∴ `3("d"^2"y")/"dx"^2 - 21"dy"/"dx" + 30"y" = 0`
∴ `("d"^2"y")/"dx"^2 - 7"dy"/"dx" + 10"y" = 0`
This is the required D.E.
Notes
The answer in the textbook is incorrect.
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