Solving Nonhomogeneous Linear Differential Equations Using Undetermined Coefficients

Given a differential equation with an initial condition $y(0) = 1$ and the differential equation $y' = 3y + 5 \sin(2t)$, find the undetermined coefficients for the particular solution and solve for $C_0$, $C_1$, and $C_2$.

In this problem, we are dealing with a first-order nonhomogeneous linear differential equation with a given initial value. The goal is to find the particular solution and subsequently solve for the constants in the complementary solution. The method of undetermined coefficients is a powerful technique used here to find the particular solution. This method is applicable when the nonhomogeneous term is a simple function such as polynomials, exponentials, sines, or cosines. The principle behind this method is to assume a form for the particular solution that is similar to the nonhomogeneous term and then determine the coefficients by substituting back into the differential equation and matching terms.

This problem begins with identifying the complementary solution to the associated homogeneous equation, followed by employing the method of undetermined coefficients to propose a form for the particular solution. Once both parts of the solution are combined—the complementary and particular solutions—initial conditions allow us to solve for any remaining arbitrary constants. This process underlines the importance of understanding the theory of linear differential equations and the techniques used to address differential equations that are not straightforwardly separable or exact. Additionally, this problem involves integrating concepts like linear independence and solutions to inhomogeneous equations, which are fundamental for more complex scenarios encountered in mechanical and electrical engineering contexts.