In the following example verify that the given expression is a solution of the corresponding differential equation: y = abxxdydxdydxa+bx;xd2ydx2+2dydx=0 - Mathematics and Statistics

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Sum

In the following example verify that the given expression is a solution of the corresponding differential equation:

y = `"a" + "b"/"x"; "x" ("d"^2"y")/"dx"^2 + 2 "dy"/"dx" = 0`

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Solution

y = `"a" + "b"/"x"`

Differentiating w.r.t. x, we get

`"dy"/"dx" = 0 + "b"(- 1/"x"^2) = - "b"/"x"^2`

∴ `"x"^2 "dy"/"dx" = - "b"`

Differentiating again w.r.t. x, we get

`"x"^2 * "d"/"dx" ("dy"/"dx") + "dy"/"dx" * "d"/"dx" ("x"^2) = 0`

∴ `"x"^2 ("d"^2"y")/"dx"^2 + "dy"/"dx" xx "2x" = 0`

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

Hence, y = `"a" + "b"/"x"` is a solution of the D.E.

`"x" ("d"^2"y")/"dx"^2 + 2 "dy"/"dx" = 0`

Concept: Formation of Differential Equations
  Is there an error in this question or solution?
Chapter 6: Differential Equations - Exercise 6.3 [Page 200]

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