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When working with differential equations, verifying that a given function is a solution involves a few crucial steps. The process starts by taking the proposed solution and calculating its derivatives according to the order of the differential equation. Once the derivatives are found, they are substituted back into the original differential equation to see if the equation holds true.

For example, consider verifying if the function
\( z(t) = C_1 e^{-t} + C_2 e^{2t} + C_3 e^{-3t} - e^t \)
is a solution to
\( z^{m}(t) + 2z''(t) - 5z'(t) - 6z(t) = 8e^{t} \).

Here, the term \( z^{m}(t) \) indicates that there will be some highest-order derivative which will be identified once the differentiations are carried out. The constants \( C_1, C_2, \) and \( C_3 \) represent any real numbers, and we need to verify if the equation is satisfied for all values of these constants.

To confirm the solution, you would compute the first and second derivatives of \( z(t) \), substitute them alongside \( z(t) \) into the left side of the differential equation, and simplify. If you end up with the right side of the equation, \( 8e^{t} \), then you have successfully verified that \( z(t) \) is a solution to the differential equation.

Calculating the first derivative of a function is fundamental in verifying solutions to differential equations. For instance, to acquire the first derivative of the function \( z(t) = C_1 e^{-t} + C_2 e^{2t} + C_3 e^{-3t} - e^t \), we apply the basic rules of differentiation to each term individually.

The differentiation of \( e^{kt} \), where \( k \) is a constant, is \( ke^{kt} \). Therefore, by differentiating each term with respect to \( t \), we get:
\( z'(t) = -C_1 e^{-t} + 2C_2 e^{2t} - 3C_3 e^{-3t} - e^t \).

This step is crucial as it provides us with one of the necessary components to plug into the original differential equation. It's important to apply the differentiation rules correctly and consistently to ensure accurate verification.

Just as with the first derivative, calculating the second derivative is a step further in the verification process of a solution to a differential equation. Continuing from where we left off with our first derivative, we differentiate it again to obtain the second derivative.

The second derivative of \( z(t) \) is found by taking the derivative of \( z'(t) \), resulting in:
\( z''(t) = C_1 e^{-t} - 4C_2 e^{2t} + 9C_3 e^{-3t} - e^t \).

Once we have the second derivative, we can substitute it, along with the original function and the first derivative, into the left side of the differential equation. The second derivative is especially important as it could reveal the presence of curvature in the graph of the function, which is essential for understanding the behavior of solutions to differential equations. In verifying our solution, it is not only about obtaining \( 8e^{t} \) but also about how the different terms involving initial function, first, and second derivatives come together and cancel out, leaving the expected result on the right side of the equation.