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The method of undetermined coefficients is a technique used to find a particular solution to a non-homogeneous linear differential equation. This method works well when the non-homogeneous term (right-hand side of the equation) is a simple function, such as a polynomial, an exponential, or a trigonometric function. The key idea is to assume a form for the particular solution that resembles the non-homogeneous term. For instance, if the non-homogeneous term is of the form \(e^{-x}\), then the guessed particular solution \(y_p\) could initially be \(Ae^{-x}\), where \(A\) is a constant to be determined.
However, if a term like \(e^{-x}\) already appears in the complementary solution, as is the case here, we adjust by incorporating a factor of \(x\) to prevent overlap, leading to the form \(y_p = Axe^{-x}\). The coefficients such as \(A\) are then determined by substituting the proposed solution into the original differential equation and solving for the unknown coefficients.

The complementary solution of a differential equation refers to the solution of the associated homogeneous equation. This type of solution specifically addresses the equation without the non-homogeneous term, allowing us to understand the natural behavior of the system without external forces.
For the given homogeneous equation \(y'' + 6y' + 9y = 0\), we solve the characteristic equation derived from this differential equation. The characteristic equation is formed as \(r^2 + 6r + 9 = 0\). On solving, it simplifies to \((r+3)^2 = 0\), indicating that \(r = -3\) is a repeated root.
In cases of repeated roots, the complementary solution is expressed as \(y_c = C_1 e^{-3x} + C_2 x e^{-3x}\), where \(C_1\) and \(C_2\) are constants determined by initial conditions. The inclusion of the term \(C_2 x e^{-3x}\) compensates for the repeated nature of the root, ensuring linear independence between terms.

Finding the particular solution to a differential equation involves addressing the non-homogeneous part of the equation. The particular solution accounts for the external inputs or forces reflected in the non-homogeneous term. In our example, the non-homogeneous component is \(e^{-x}\). By employing the method of undetermined coefficients, we proposed a particular solution of the form \(y_p = Axe^{-x}\), suitably adjusted to accommodate repetition in the complementary function.
To establish the particular solution, substitute \(y_p\) along with its derivatives back into the original differential equation to determine the unknown coefficients. Simplifying and equating coefficients yields the constant \(A = \frac{1}{4}\). This gives us the particular solution \(y_p = \frac{1}{4}xe^{-x}\).
Finally, the general solution of the differential equation integrates both the complementary and particular solutions, written as \(y = C_1 e^{-3x} + C_2 x e^{-3x} + \frac{1}{4}xe^{-x}\). This comprehensive solution encompasses both natural and forced responses of the system.