The method of undetermined coefficients is used to find a particular solution to a non-homogeneous linear differential equation. It works best when \( g(x) \) is a simple function, such as polynomials, exponentials, sines and cosines, or their finite sums/products.
The process involves:
- Guessing the form of the particular solution based on \( g(x) \)'s form.
- Substituting this guess into the original differential equation.
- Solving for the unknown coefficients by equating terms on both sides of the equation.
For our example equation, \( y'' - 2y' + 5y = 5x + 4e^x(1+\text{sin}(2x)) \), the trial solution will be: \[ y_p = Ax + B + e^x(C + Dx \text{sin}(2x) + Ex \text{cos}(2x)) \].
Here we break it down:
- The term \( 5x \) suggests a guess like \( Ax + B \).
- The term \( 4e^x(1+\text{sin}(2x)) \) suggests a guess like \( e^x(C + Dx \text{sin}(2x) + Ex \text{cos}(2x)) \).
Using this combined form, substituting into the original equation will help us determine the coefficients \( A, B, C, D, \) and \( E \).