Answer:
Step-by-step explanation:
This is a <em>homogeneous linear third-order differential equation</em>
z′′′+2z′′−4z′−8z=0 so the
FIRST STEP is to find the characteristic equation and its roots
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Using the method of finding roots of a polynomial (using m = 2) would provide the solution below
(m-2)(m+2)(m+2)=0; m = 2 and m = -2 (repeated twice)
SECOND STEP is to write the general solution with the arbitrary constants.
The general solution based on the roots and the using x as the independent variable would provide
<em>*It should be noted that when the characteristic equation has a repeated root, the general equation form becomes similar to the last part of the answer*</em>