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Question
Obtain the differential equation by eliminating the arbitrary constants from the following equation:
y = a + `"a"/"x"`
Solution
y = a + `"a"/"x"` ....(1)
Differentiating twice w.r.t. x, we get
`"dy"/"dx" = "d"/"dx"("a" + "a"/"x") = 0 + "a"(- 1/"x"^2)`
∴ `"dy"/"dx" = - "a"/"x"^2`
∴ `"a" = - "x"^2 "dy"/"dx"`
Substituting the value of a in (1), we get
y = - `"x"^2 "dy"/"dx" + 1/"x"(- "x"^2 "dy"/"dx")`
∴ y = -`"x"^2 "dy"/"dx" - "x" "dy"/"dx"`
∴ `("x"^2 + "x") "dy"/"dx" + "y" = 0`
∴ x(x + 1) `"dy"/"dx" + "y" = 0`
This is the required D.E.
Notes
The answer in the textbook is incorrect.
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