91Ó°ÊÓ

The characteristic equation is crucial when solving linear homogeneous differential equations. It's formed by replacing derivatives in the equation with powers of a variable, typically denoted as \(r\). This step converts the differential equation into an algebraic one, making it easier to handle.
For example, consider the differential equation \(y^{\prime \prime}-3 y^{\prime}+8 y=0\). You first rewrite it in the standard form \(ay^{\prime\prime} + by^{\prime} + cy = 0\). Identifying the values, we substitute \(a = 1\), \(b = -3\), and \(c = 8\).
The characteristic equation becomes \(ar^2 + br + c = 0\) or \(r^2 - 3r + 8 = 0\). Solving this quadratic equation will allow us to find the roots, helping us identify the type of solutions for the differential equation.

In the context of differential equations, complex roots often occur when the discriminant of the characteristic equation is negative. This happens, as seen in the equation \(r^2 - 3r + 8 = 0\), where the discriminant \(b^2 - 4ac\) evaluates to \(-23\), a negative number.
For complex roots, we use the quadratic formula \(r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), which results in imaginary components. In our example:

• \(r_1 = \frac{3 + \sqrt{-23}}{2}\)
• \(r_2 = \frac{3 - \sqrt{-23}}{2}\)

These roots are expressed as \(\alpha \pm \beta i\), where \(\alpha\) is the real part and \(\beta\) is derived from the imaginary part. This form helps us to construct the general solution for the differential equation.

A homogeneous linear differential equation, like \(y^{\prime \prime}-3 y^{\prime}+8 y=0\), has solutions that can be superimposed to form other solutions. This property makes it particularly useful in various practical fields, including physics and engineering.
To solve these equations with complex roots, you derive a general solution of the form:

• \(y(x) = e^{\alpha x}(C_1\cos(\beta x) + C_2\sin(\beta x))\)

Here, \(\alpha\) is the real part of the root, while \(\beta\) pertains to the imaginary part. For our example:

• \(\alpha = \frac{3}{2}\)
• \(\beta = \frac{\sqrt{-23}}{2}\)

The constants \(C_1\) and \(C_2\) are determined from initial conditions. The result is a solution that describes oscillatory motion modulated by exponential growth or decay, essential for modeling real-world phenomena.