Final Answer:
The provided solution for x'' + ω²x = 0 with initial values x(0) = x₀ and x'(0) = u₀ is x₀e^t - (x₀ - u₀)(1 - e^(-t)).
Step-by-step explanation:
The differential equation x'' + ω²x = 0 represents a simple harmonic oscillator. This second-order linear differential equation has a characteristic equation r² + ω² = 0, where r is a complex number. The general solution for this type of equation is x(t) = Acos(ωt) + Bsin(ωt).
In this case, the initial conditions x(0) = x₀ and x'(0) = u₀ are used to solve for the constants A and B. The solution x₀e^t - (x₀ - u₀)(1 - e^(-t)) is derived from the initial value problem by applying these initial conditions to the general solution.
Breaking down the provided solution:
x₀e^t represents the homogeneous solution, satisfying the equation x'' + ω²x = 0. It corresponds to the particular solution of the differential equation with x(0) = x₀.
-(x₀ - u₀)(1 - e^(-t)) is the complementary solution, ensuring that when t = 0, it evaluates to x'(0) = u₀.
The solution combines these two components to satisfy both initial conditions. It's noteworthy that this form of the solution is often derived through the method of undetermined coefficients or by using the method of variation of parameters to solve the given differential equation with initial conditions.