Solution 8.

Use Mathematica's subroutine  DSolve to find the solution.

Find the roots of the characteristic equation.

Form the general solution using the roots of the characteristic equation.

Verify that f[x] and its derivatives satisfy the D. E.

Remark the general solution using real functions.
This can be accomplished by considering the complex pair.

Look at their real and imaginary parts.

The two linearly independent real functions are:

Use them in forming a new general solution.

Verify that f[x] and its derivatives satisfy the D. E.