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Question
For the given below, verify that the given function (implicit or explicit) is a solution to the corresponding differential equation.
xy = a ex + b e-x + x2 : `x (d^2y)/(dx^2) + 2 dy/dx - xy + x^2 - 2 = 0`
Solution
Given function xy = a ex + b e-x + x2
On differentiating with respect to x,
`dy/dx = ae^x - be^-x + 2x`
On differentiating again,
`(d^2y)/dx^2 = ae^x + be^-x + 2`
L.H.S. ⇒ `x (d^2y)/(dx^2) + 2 dy/dx - xy + x^2 - 2`
⇒ x(aex + be-x + 2) + 2(aex - be-x + 2x) - x(aex + be-x + x2) + x2 - 2
⇒ ex (ax + 2a - ax) + e-x (bx - 2b - bx) + 2x + 4x - x3 + x2 - 2
`= 2ae^x - 2be^(- x) - x^3 + x^2 - 2 ne 0`
Hence, L.H.S. ≠ R.H.S.
Hence, the given function is not a solution of the differential equation.
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