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Question
In the following example verify that the given expression is a solution of the corresponding differential equation:
y = e-x + Ax + B; `"e"^"x" ("d"^2"y")/"dx"^2 = 1`
Solution
y = e-x + Ax + B;
Differentiating w.r.t. x, we get
`"dy"/"dx" = "e"^-"x" xx (- 1) + "A" xx 1 + 0`
`∴ "dy"/"dx" = "e"^-"x" + "A"`
Differentiating again w.r.t. x, we get
`("d"^2"y")/"dx"^2 = - "e"^-"x" xx (- 1) + 0`
∴ `("d"^2"y")/"dx"^2 = 1/"e"^"x"`
∴ `"e"^"x" ("d"^2"y")/"dx"^2 = 1`
Hence, y = e-x + Ax + B is a solution of the D.E.
`"e"^"x" ("d"^2"y")/"dx"^2 = 1`
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