It has to be shown that `y1 = e^(-3x) and y2 = e^(4x)` are the solutions of `(d^2y)/dx^2 - dy/dx - 12*y = 0`

y = e^-3x

=> dy/dx = -3*e^-3x and `(d^2y)/dx^2 = 9*e^(-3x)`

`(d^2y)/dx^2 - dy/dx - 12y`

=> `9*e^(-3x) + 3*e^(-3x) - 12*e^(-3x)`

=> 0

Similarly, if...

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It has to be shown that `y1 = e^(-3x) and y2 = e^(4x)` are the solutions of `(d^2y)/dx^2 - dy/dx - 12*y = 0`

y = e^-3x

=> dy/dx = -3*e^-3x and `(d^2y)/dx^2 = 9*e^(-3x)`

`(d^2y)/dx^2 - dy/dx - 12y`

=> `9*e^(-3x) + 3*e^(-3x) - 12*e^(-3x)`

=> 0

Similarly, if y = e^4x

`dy/dx =4*e^(4x)`

`(d^2y)/(dx^2) = 16*e^(4x)`

`(d^2y)/dx^2 - dy/dx - 12y`

=> `16*e^(4x) - 4*e^(4x) - 12*e^(4x)`

=> 0

This proves that `y1 = e^(-3x) and y2 = e^(4x)` are the solutions of `(d^2y)/dx^2 - dy/dx - 12y = 0`