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When we deal with second-order linear differential equations like \( y^{\prime \prime} - y^{\prime} - 2y = 6 \), identifying the characteristic equation is a critical step, especially when solving the homogeneous part of the equation. The characteristic equation stems from assuming a solution of the form \( y = e^{rt} \) for the homogeneous equation \( y^{\prime \prime} - y^{\prime} - 2y = 0 \).

This assumption transforms the differential equation into an algebraic one. By plugging \( y = e^{rt} \) into the homogeneous equation and simplifying, we find:

• \( y^{\prime \prime} = r^2 e^{rt} \)
• \( y^{\prime} = r e^{rt} \)
• Which simplifies the expression to : \( r^2 e^{rt} - r e^{rt} - 2 e^{rt} = 0 \)

By factoring out \( e^{rt} \), which is never zero, we are left with the characteristic equation: \( r^2 - r - 2 = 0 \). Solving this quadratic provides the roots that determine the form of the solution to the homogeneous equation.

The method of undetermined coefficients is a clever way to find particular solutions to non-homogeneous differential equations. This technique relies on educated guessing based on the form of the non-homogeneous term, allowing the solution process to be more strategic and less cumbersome.

For the equation \( y^{\prime \prime} - y^{\prime} - 2y = 6 \), the non-homogeneous term is a constant (6). Hence, we start by guessing that a particular solution, \( y_p \), could also be a constant. We write \( y_p = A \) where \( A \) is a constant.

When we differentiate \( y_p \), the derivatives \( y_p^{\prime} = 0 \) and \( y_p^{\prime\prime} = 0 \) yield zero, simplifying substitutions in our main equation. Substituting \( y_p \) into the original equation gives us \( -2A = 6 \). Solving this, we find \( A = -3 \), and thus the particular solution is \( y_p = -3 \).

This method effectively narrows down possible solutions, utilizing trial and error within a structured guessing approach, saving time and computation.

The general solution of a non-homogeneous differential equation combines both the solutions to its homogeneous and particular parts. Once we have solved:

• The homogeneous equation \( y_h = C_1 e^{2t} + C_2 e^{-t} \)
• The particular solution \( y_p = -3 \)

We can construct the general solution by adding these two components together. Thus, the general solution of the equation \( y^{\prime \prime} - y^{\prime} - 2y = 6 \) is:

\[ y = y_h + y_p = C_1 e^{2t} + C_2 e^{-t} - 3 \]

Here,

• \( C_1 \) and \( C_2 \) are arbitrary constants. These can be determined given initial conditions.

This solution effectively captures all possible behaviors of the differential equation, due to the combination of exponential growth/decay from the homogeneous solution and the steady-state offset from the particular solution.

A differential equation is termed homogeneous when there are no non-zero terms independent of the solution function, leading to the expression on the right-hand side equalling zero. For our example, the homogeneous equation is \( y^{\prime\prime} - y^{\prime} - 2y = 0 \).

The importance of solving this equation lies in establishing a fundamental set of solutions, which, when summed, form the homogeneous solution, \( y_h \).

To solve it, we:

• Assume a solution of the form \( y = e^{rt} \).
• Derive the characteristic equation \( r^2 - r - 2 = 0 \).
• Solve for \( r \), yielding roots \( r = 2 \) and \( r = -1 \).

With distinct real roots, the general solution for this homogeneous differential equation is the linear combination:

\[ y_h = C_1 e^{2t} + C_2 e^{-t} \]

This represents all behaviors dictated by the linear homogeneous nature of the equation, such as exponential growth or decay.