Solving Differential Equations with Undetermined Coefficients: A Comprehensive Guide

Introduction

Differential equations are a critical part of mathematical modeling in various fields, including engineering, physics, and chemistry. Among different methods to solve differential equations, the method of undetermined coefficients is particularly useful for equations with non-homogeneous parts, such as the term 2e^2x cos(x).

Solving the Given Differential Equation

Consider the differential equation:

y'' - 4y' 5y 2e^2x cos(x)

We will solve this step-by-step, starting with the homogeneous equation and then finding the particular solution.

1. Solving the Homogeneous Equation

The homogeneous part of the given equation is:

y'' - 4y' 5y 0

To solve this, we assume a solution of the form y e^(λx). Substituting this into the equation, we get:

λ^2 e^(λx) - 4λ e^(λx) 5e^(λx) 0

Factoring out e^(λx), we get the characteristic equation:

λ^2 - 4λ 5 0

Solving for λ, we get:

λ 2 ± i

This gives us the complementary solution:

y_c(x) c_1 e^(2x) cos(x) c_2 e^(2x) sin(x)

2. Finding the Particular Solution

For the particular solution, we assume:

y_p(x) a_1 e^(2x) cos(x) a_2 e^(2x) sin(x)

We differentiate y_p(x) twice:

y_p'(x) (2a_1 a_2) e^(2x) cos(x) (a_1 - 2a_2) e^(2x) sin(x)

y_p''(x) (4a_1 4a_2 - 4) e^(2x) cos(x) (4a_1 - 4a_2 4) e^(2x) sin(x)

Substituting these into the original differential equation:

(4a_1 4a_2 - 4) e^(2x) cos(x) (4a_1 - 4a_2 4) e^(2x) sin(x) - 4 (2a_1 a_2) e^(2x) cos(x) - 4 (a_1 - 2a_2) e^(2x) sin(x) 5 (a_1 e^(2x) cos(x) a_2 e^(2x) sin(x)) 2e^2x cos(x)

Simplifying and equating the coefficients of e^(2x) cos(x) and e^(2x) sin(x), we get:

a_1 -e^2/4, a_2 -e^2/4, a_3 e^2/8, a_4 e^2/4

Therefore, the particular solution is:

y_p(x) -1/4 e^2 cos(x) - 1/4 e^2 x cos(x) - 1/4 e^2 sin(x) - 1/4 e^2 x sin(x)

Simplifying further:

y_p(x) 1/8 e^2 cos(x) - 1/4 e^2 x cos(x) - 1/4 e^2 sin(x) - 1/4 e^2 x sin(x)

3. General Solution

The general solution is the sum of the complementary and particular solutions:

y(x) c_1 e^(2x) cos(x) c_2 e^(2x) sin(x) 1/8 e^2 cos(x) - 1/4 e^2 x cos(x) - 1/4 e^2 sin(x) - 1/4 e^2 x sin(x)

This is the complete solution to the given differential equation. The constants c_1, c_2 are determined by initial or boundary conditions.

Conclusion

The method of undetermined coefficients is a powerful tool for solving inhomogeneous linear differential equations. It provides a systematic approach to finding particular solutions and combines them with the homogeneous solution to give the overall solution. This method is widely used in various scientific and engineering applications.