Ans the following question plzzzzzzzzzz... Show that Y1=e-3x and Y2=e4x are the solutions of the following differential equation. Also find the general solution of it. Moreover, shows that Y1=e-3x and Y2=e4x form a fundamental set of solutions on (-∞,∞) d2y/dx2 – dy/dx – 12y = 0

You need to solve the second order homogeneous differential equation, hence, you need to write the characteristic equation, such that:

`(d^2y)/(dx^2) = k^2`

`(dy)/(dx) = k`

`k^2 - k - 12 = 0`

You need to use the quadratic formula, such that:

`k_(1,2) = (1+-sqrt(1 + 48))/2 => k_(1,2) =...

You need to solve the second order homogeneous differential equation, hence, you need to write the characteristic equation, such that:

`(d^2y)/(dx^2) = k^2`

`(dy)/(dx) = k`

`k^2 - k - 12 = 0`

You need to use the quadratic formula, such that:

`k_(1,2) = (1+-sqrt(1 + 48))/2 => k_(1,2) = (1+-sqrt49)/2`

`k_1 = 4; k_22 = -3`

Since the solutions to characteristic equation are real and different yields the general solution to the differential equation, such that:

`y = c_1*e^(k_1x) + c_2*e^(k_2x)`

`y = c_1*e^(4x) + c_2*e^(-3x)`

Hence, evaluating the general solution to the given differential equation, yields `y = c_1*e^(4x) + c_2*e^(-3x).`

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