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Show that y=a*e^(2x)+b*e^(-x) is a solut...

Show that `y=a*e^(2x)+b*e^(-x)` is a solution of the differential equation `(d^(2)y)/(dx^(2))-(dy)/(dx)-2y=0`.

Text Solution

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`y=ae^(2x) +be ^(-x).............(1)`
`(dy)/(dx)=2ae^(2x)-be^(-x).............(2)`
`(d^2y)/(dx^2)=4ae^(2x)+be^(-x).............(3)`
putting values on LHS
`=(d^2y)/(dx^2)-(dy)/(dx)-2y`
`=(4ae^(2x)+be^(-x))-(2ae^(2x)-be^(-x))-2(ae^(2x)+be^(-x))`
`=4ae^(2x) +be^(-x)-2ae^(2x)+be^(-x)-2ae^(2x)-2be^(-x)`
`=0`
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