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When dealing with second-order linear homogeneous differential equations like \( y'' + 6y' + 13y = 0 \), the characteristic equation is a pivotal element. It's a crucial tool used to determine the behavior of solutions. For any second-order linear differential equation of the form \( ay'' + by' + cy = 0 \), its characteristic equation is obtained by replacing the derivatives with powers of \( r \):

• Replace \( y'' \) with \( r^2 \).
• Replace \( y' \) with \( r \).
• Replace \( y \) with \( 1 \).

For our example, this gives the quadratic \( r^2 + 6r + 13 = 0 \). This quadratic equation helps us understand the nature of the roots, which in turn, influences the form of the general solution.

The characteristic equation \( r^2 + 6r + 13 = 0 \) has its roots calculated using the quadratic formula: \[ r_{1,2} = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] Here, \( a = 1 \), \( b = 6 \), and \( c = 13 \). When we input these values, we find: \[ r = \frac{-6 \pm \sqrt{36 - 52}}{2} = \frac{-6 \pm \sqrt{-16}}{2} \] Given \( \sqrt{-16} = 4i \), this results in the complex roots:

• \( r_1 = -3 + 2i \)
• \( r_2 = -3 - 2i \)

Complex roots in differential equations imply oscillatory solutions that decay over time, due to the exponential component derived from the real part of the roots. This contributes significantly to the form of the general solution.

Once we have determined that our characteristic equation yields complex roots, the general solution for the differential equation is crafted to represent both the oscillatory and exponential decay behavior of the solutions. For roots \( \alpha \pm \beta i \) (where \( \alpha = -3 \) and \( \beta = 2 \) in our case), the general solution takes the form:\[ y(t) = e^{\alpha t}\(C_1 \cos(\beta t) + C_2 \sin(\beta t)\) \] Thus, for our specific example:\[ y(t) = e^{-3t}(C_1 \cos(2t) + C_2 \sin(2t)) \] This solution format displays how the exponential decay term \( e^{-3t} \) modulates the amplitude of the oscillatory cosine and sine components, reflecting the characteristics of the differential equation's roots and providing a complete description of its behavior.