Nonhomogeneous Second Order ODE with Exponential Forcing Function

Solve the non-homogeneous ordinary differential equation: $y'' - 2y' - 3y = 3e^{2t}$ using the method of undetermined coefficients.

In solving the given non-homogeneous ordinary differential equation (ODE) using the method of undetermined coefficients, one starts by understanding the key strategy: decompose the solution into complementary and particular parts. The complementary solution is derived from the corresponding homogeneous equation by considering the characteristic equation. This step reveals the natural response of the system without external forces, and typically involves solving a polynomial for its roots to understand the behavior of the solution; whether it is decaying, oscillatory, or stationary.

The particular solution, on the other hand, caters to the non-homogeneous part of the equation, which in this case is an exponential function. Here, the method of undetermined coefficients is particularly effective when the non-homogeneous term falls into a certain form, such as polynomials, exponentials, or sine and cosine functions. By hypothesizing a form of the solution—a function with unspecified coefficients—you can deduce these coefficients by substituting back into the original equation and equating terms. This provides insight into how external forces affect the system beyond its natural behavior.

Together, these components form the general solution to the ODE, offering a comprehensive view into both inherent and externally driven dynamics. This process underscores the importance of recognizing the composition of solutions in differential equations and choosing an appropriate method based on the form of non-homogeneity encountered in physical and mathematical problems.