Final answer:
To solve the initial value problem y'' + 5y = 0, we assume a solution of the form y = e^(rx) and find the roots of the characteristic equation. Since the roots are complex, the general solution is y = Ae^(√5x)i + Be^(-√5x)i.
Step-by-step explanation:
The given equation is a second-order linear homogeneous differential equation. To solve this equation, we assume a solution of the form y = e^(rx), where r is a constant. Substituting this into the equation, we get the characteristic equation r^2 + 5 = 0. Solving this equation gives us r = ±√(-5). Since the discriminant is negative, the roots are complex. Therefore, the general solution to the equation is y = Ae^(√5x)i + Be^(-√5x)i, where A and B are constants.