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Question
Find the differential equation corresponding to the family of curves represented by the equation y = Ae8x + Be –8x, where A and B are arbitrary constants
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Solution
Given y = Ae8x + Be–8x .........(1)
Where A and B are arbitrary constants differentiating equation (1) twice successively because we have two arbitrary constant, we get
`("d"y)/("d"x)` = Ae8x + Be–8x (– 8)
`("d"y)/("d"x)` = 8Ae8x – 8Be–8x ........(2)
`("d"^2y)/("d"x^2)` = 8Ae8x (8) – Be–8x (– 8)
= 64Ae8x + 64Be–8x
= 64[Ae8x + Be–8x] .........(3)
Substituting equation (1) in eqn (3), we get
`("d"^2y)/("d"x^2)` = 64y
`("d"^2y)/("d"x^2) - 64y` = 0 is the required differential equation.
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