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The characteristic equation is a vital concept in solving second-order linear differential equations with constant coefficients. It helps to determine the behavior of solutions to the differential equation. When you have a differential equation of the form \( y'' + ay' + by = 0 \), you convert it into a characteristic equation by replacing each derivative with powers of \( r \). Here's the step-by-step process:

• Replace \( y'' \) with \( r^2 \).
• Replace \( y' \) with \( r \).
• Leave \( y \) as a constant 1, since it corresponds to \( r^0 \).

That turns the differential equation into \( r^2 + ar + b = 0 \). In our example, this gives us \( r^2 + 4r + 5 = 0 \). By solving this quadratic equation, we find the roots, which are crucial for forming the general solution of the differential equation.

When solving the characteristic equation, the nature of its roots decides the type of solutions we will get for the differential equation. The discriminant gives us insight:

• If the discriminant \( b^2 - 4ac \) is positive, we have real roots.
• If it's zero, we have repeated real roots.
• If it's negative, the roots are complex.

Complex roots appear in conjugate pairs, like \( \alpha \pm \beta i \). In this specific case, our characteristic equation is \( r^2 + 4r + 5 = 0 \), and the discriminant, \( 16 - 20 \), equals \( -4 \). Since the discriminant is negative, we solve for roots using:\[ r = \frac{-4 \pm \sqrt{-4}}{2} = -2 \pm i \]This indicates that the roots are complex, \( -2 + i \) and \( -2 - i \), leading to an oscillatory component in the general solution.

The general solution of a differential equation with complex roots combines exponential and trigonometric functions. This structural form results from Euler's formula, which connects complex exponentials to sine and cosine functions. For complex roots \( \alpha \pm \beta i \), the general solution is:\[ y(t) = e^{\alpha t}(C_1 \cos(\beta t) + C_2 \sin(\beta t)) \]In our example:

• \( \alpha = -2 \)
• \( \beta = 1 \)

Thus, the general solution is expressed as:\[ y(t) = e^{-2t}(C_1 \cos(t) + C_2 \sin(t)) \]This tells us the solution will decay over time (due to \( e^{-2t} \)), with oscillations determined by the cosine and sine terms. The constants \( C_1 \) and \( C_2 \) are found using initial conditions, representing the particular solution fitting specific situations.