Final Answer:
The solution to the given second-order differential equation is y(x) = 2e^(-7x) + e^(-3x).
Step-by-step explanation:
To solve the given second-order differential equation d²y/dx² + 10dy/dx + 21y = 0, we can start by finding the characteristic equation. The characteristic equation for a second-order linear homogeneous differential equation is given by r² + br + c = 0, where b and c are the coefficients of the first and zeroth order terms, respectively. In this case, the characteristic equation is r² + 10r + 21 = 0. Solving for r, we get the roots r₁ = -3 and r₂ = -7.
The general solution to the given differential equation is y(x) = C₁e^(r₁x) + C₂e^(r₂x), where C₁ and C₂ are constants to be determined using the initial conditions. Applying the initial conditions y(0) = 2 and dy/dx(0) = 2, we can find the specific values of C₁ and C₂. Substituting these values into the general solution yields the particular solution y(x) = 2e^(-7x) + e^(-3x).
Therefore, the final answer to the given second-order differential equation is y(x) = 2e^(-7x) + e^(-3x), satisfying both the differential equation and the initial conditions.