Final answer:
The solution to the initial value problem y'' + 4y' + 13y = 0 involves finding the characteristic equation and solving for the complex roots. The general solution is a linear combination of exponential functions.
Step-by-step explanation:
The given equation is a linear homogeneous second-order differential equation: y'' + 4y' + 13y = 0. To find the solution to this initial value problem, we can assume a solution of the form y = e^(rt), where r is a complex number. By substituting this into the equation, we can find the characteristic equation and solve for r. The general solution will then be y = c₁e^(r₁t) + c₂e^(r₂t), where c₁ and c₂ are constants determined by the initial conditions.