Solving Differential Equations Using Laplace Transforms2

Solve the differential equation $y' + 4y = e^{4t}$ using Laplace transforms, given the initial condition $y(0) = 3$.

In solving this differential equation using Laplace transforms, it's important to understand the role of the Laplace transform in transforming differential equations into algebraic ones. This step simplifies the process of solving the equation by making the solution of differential equations systematic. Here, the Laplace transform will convert the given linear differential equation into an algebraic equation in terms of the Laplace variable. We will then solve for the Laplace transform of the unknown function.

Once the algebraic equation is obtained, we can solve for the Laplace transform of the solution, which involves algebraic manipulation and the application of inverse Laplace transforms. These processes highlight the importance of understanding various pairs of functions and their transforms, especially exponential and unit step functions. In this problem, recognizing the function involved and its behavior under Laplace transformations is crucial.

The initial condition provides a boundary value that is essential in determining the particular solution to the problem. This problem emphasizes the importance of initial conditions in obtaining the unique solution to differential equations using Laplace transforms. This method is a powerful tool for solving linear differential equations with constant coefficients, especially when dealing with nonhomogeneous terms such as exponentials.