Solve the given initial-value problem.

To solve the initial-value problem given by the differential equation

we start by rewriting the equation as follows:

This is a third-order linear homogeneous differential equation. We can find its characteristic equation by assuming a solution of the form , which leads to:

Factoring out an , we have:

This gives us one root . For the quadratic part , we can factor it as:

yielding a double root . Therefore, the general solution for the differential equation is:

To find , , and , we need to use the initial conditions: , , and .

Calculating :

Now, let’s calculate :

Evaluating :

Substituting :

Next, we calculate :

Evaluating :

Substituting and :

Thus, and .

Finally, substituting the values of , , and back into the general solution, we have:

Thus, the final solution to the initial-value problem is: