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Differential equations are mathematical equations that relate a function with its derivatives. They are used to describe various physical phenomena, such as heat transfer, wave propagation, and population dynamics. In essence, a differential equation specifies how a quantity changes over time or space. Differential equations are classified based on their order, linearity, and homogeneity.

For instance, the given equation in our problem, \( y'' - y = 2 \tanh x \), is a second-order linear differential equation. Here's why:

• **Second-order:** The highest derivative in the equation is the second derivative, \( y'' \).
• **Linear:** Each term involving the function \( y \) or its derivatives is linear, meaning it is not raised to a power or multiplied by another derivative.
• **Inhomogeneous:** The equation includes a nonzero term on the right-hand side, \( 2 \tanh x \), making it inhomogeneous.

Understanding the nature of the differential equation is crucial because it dictates the methods and techniques available for finding solutions, such as the variation of parameters method used in this exercise.

The complementary function is a critical part of solving non-homogeneous linear differential equations. It satisfies the corresponding homogeneous equation, where all non-homogeneous terms are set to zero. For the equation \( y'' - y = 0 \), the complementary function accounts for the solution of the associated homogeneous differential equation.

In our exercise, finding the complementary function involves determining the roots of the characteristic equation \( r^2 - 1 = 0 \). Solving this, we obtain roots \( r = 1 \) and \( r = -1 \). The complementary function, denoted as \( y_c(x) \), is then constructed as follows:

• \( y_c(x) = c_1e^x + c_2e^{-x} \)

Here, \( c_1 \) and \( c_2 \) are arbitrary constants that will be determined once initial or boundary conditions are applied. The complementary function captures all possible solutions to the homogeneous portion of the differential equation. This step sets the stage for finding the particular solution by tackling the non-homogeneous part next.

The particular solution of a non-homogeneous differential equation addresses the specific terms on the right-hand side, independent of arbitrary constants. In our problem, the challenge is to account for the term \( 2 \tanh x \). The variation of parameters method is a technique employed here to determine a particular solution.

This method involves first expressing the particular solution \( y_p(x) \) as a linear combination of the homogeneous solutions with variable coefficients:

• \( y_p(x) = u_1(x)e^x + u_2(x)e^{-x} \)

Following the procedure means deriving expressions and simplifying to isolate the terms \( u_1'(x) \) and \( u_2'(x) \). These derivatives are then integrated to find \( u_1(x) \) and \( u_2(x) \). The complete particular solution is derived by substituting these functions back into the expression for \( y_p(x) \). Importantly, the particular solution captures the behavior defined solely by the non-homogeneous term \( 2 \tanh x \). This solution will combine appropriately with the complementary function to give a comprehensive depiction of the system described by the differential equation.

The general solution of a differential equation combines the complementary function and the particular solution, accounting for both the homogeneous and non-homogeneous parts of the equation. It signifies the total set of solutions for the equation.

For the differential equation given, the general solution is composed of:

• The complementary function \( y_c(x) = c_1e^x + c_2e^{-x} \)
• The particular solution derived from the non-homogeneous component \( y_p(x) = \frac{-k}{2e^{x}} + \frac{l}{2e^{-x}} \)

Thus, the general solution is expressed as:\[ y(x) = c_1e^x + c_2e^{-x} + \frac{-k}{2e^{x}} + \frac{l}{2e^{-x}} \]Within this solution, \( c_1 \) and \( c_2 \) are arbitrary constants, capturing the infinite solutions defined by different initial conditions. The terms involving \( k \) and \( l \) stem from integrating the particular solution. The general solution fully characterizes the behavior of the differential equation over its domain.