Final answer:
To solve the differential equation, we first find the roots of the characteristic equation. The general solution is a combination of the homogeneous and particular solutions.
Step-by-step explanation:
To find the general solution of the differential equation x'' + 3x' + (1/8 + 10)x = 10 with initial conditions x(0) = -2 and x'(0) = 0, we can first solve the homogeneous equation by finding the roots of the characteristic equation. The characteristic equation is r^2 + 3r + 81/8 = 0. Solving this equation gives us two complex roots: r = -3/2 + (3/2)i and r = -3/2 - (3/2)i.
Since we have complex roots, the general solution of the homogeneous equation is x(t) = e^(-3t/2)(C1cos((3√3/2)t) + C2sin((3√3/2)t)).
To obtain the particular solution, we assume x(t) = A. Substituting this into the original equation, we find A = 7/12.
Therefore, the general solution of the given differential equation is x(t) = e^(-3t/2)(C1cos((3√3/2)t) + C2sin((3√3/2)t)) + 7/12.