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A characteristic equation is a crucial concept when solving differential equations, especially second-order linear ones like the given example. This equation often emerges from a method that proposes solutions of the form \( y = e^{rt} \). By substituting this into the differential equation, we transform the problem of solving a differential equation into solving a polynomial equation, specifically a quadratic equation for second-order cases.
This polynomial, known as the characteristic equation, is found by replacing derivatives in the differential equation with powers of \( r \), as was done in the problem: \( 4r^2 - 4r + 1 = 0 \).

• The coefficients in the differential equation determine the coefficients of the characteristic equation.
• Solving the characteristic equation yields the roots \( r \), which are used to construct the general solution of the differential equation.

Understanding how to derive the characteristic equation and solve it is key to tackling second-order linear differential equations effectively.

Repeated roots occur when solving the characteristic equation yields the same value for \( r \) more than once. In the context of second-order linear differential equations, a repeated root presents a unique scenario.
In this exercise, the characteristic equation \( 4r^2 - 4r + 1 = 0 \) simplifies to a single repeated root, \( r = \frac{1}{2} \).

• Repeated roots occur when the discriminant \( b^2 - 4ac \) of the quadratic formula is zero, leading to a single solution being repeated.
• This means instead of two distinct solutions, we obtain one solution repeated, which requires special handling in terms of forming the general solution.

Recognizing repeated roots is essential because it alters the form of the general solution, incorporating terms to account for the multiplicity of the root.

The general solution to a differential equation encompasses all possible solutions. For a second-order linear homogeneous differential equation with repeated roots, this solution takes a distinctive form.
In cases like \( r = \frac{1}{2} \) repeated, the solution is not just \( e^{rac{1}{2}x} \). We use the formula for repeated roots, which adds a term involving \( x \):
\[ y(x) = (C_1 + C_2 x) e^{rac{1}{2}x} \]

• \( C_1 \) and \( C_2 \) are arbitrary constants, showing the infinite nature of solutions as \( x \) varies.
• The term \( C_2 x \) accounts for the repeated nature of the root, ensuring completeness in capturing all potential solutions.

Understanding how to construct the general solution, especially in the context of repeated roots, is essential for solving the differential equation correctly and comprehensively.