Solve Initial Value Problem using Laplace Transform

Consider the differential equation $y'' - 2y' + 5y = 0$ with initial conditions $y(0) = 2$ and $y'(0) = -4$. Solve the initial value problem using the Laplace transform.

The process of solving differential equations using the Laplace transform provides an effective method for dealing with initial value problems, particularly when such problems involve linear equations with constant coefficients. The Laplace transform converts a differential equation in the time domain into an algebraic equation in the Laplace domain, where it is generally easier to manipulate and solve. Once the algebraic form is solved for the Laplace transform of the solution, the inverse transform is used to convert the solution back into the time domain. This method is particularly advantageous as it handles complicated forcing functions easily and helps in solving systems with initial conditions without needing to find a general solution first.

Exploring the initial value problem with the given differential equation, the Laplace transform provides a systematic approach to include initial conditions directly into the transformation process. The initial conditions are incorporated when transforming the derivatives, helping to streamline the solving process. Studying such problems emphasizes understanding the transformation rules and inverse transformations. The problem also enhances comprehension of differential equations and the practical application of Laplace transforms in engineering and physics to solve real-world linear differential equations with initial values. This approach is key for simplifying complex systems and predicting long-term behavior of systems accurately.