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Question
Obtain the differential equation by eliminating the arbitrary constants from the following equation :
`y = c_1e^(2x) + c_2e^(-2x)`
Solution
`y = c_1e^(2x) + c_2e^(-2x)`
differentiate w.r.t. x.
`(dy)/(dx) = 2c_1e^(2x) - 2c_2e^(-2x)`
Again diff. w.r.t. x.
`(d^2y)/(dx^2) =4c_1e^(2x) + 4c_2e^(-2x)`
`= 4(c_1e^(2x) + c_2e^(-2x))`
= 4y
`:. (d^2y)/(dx^2) - 4y = 0`
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