91Ó°ÊÓ

When solving second-order linear homogeneous differential equations, the characteristic equation is a crucial tool. For equations of the form \( ay'' + by' + cy = 0 \), we assume a trial solution \( y = e^{rt} \). By substituting this back into the differential equation, we transform it into an algebraic equation in terms of \( r \), known as the characteristic equation. In our exercise example, the characteristic equation we formed from \( 4y'' + 20y = 0 \) was \( 4r^2 + 20 = 0 \). This helps us find the roots, which guide us towards the general solution of the differential equation.

In algebra, a quadratic equation like \( ar^2 + br + c = 0 \) might yield complex roots, especially when the discriminant \( b^2 - 4ac \) is negative. In such a case, the solutions take the form \( r = \alpha \pm i\beta \). In our exercise, simplifying the characteristic equation gave us \( r^2 = -5 \), leading to complex roots \( r = \pm i\sqrt{5} \).
The presence of complex roots indicates oscillatory solutions in the time domain. For our differential equation, this oscillation is represented by sine and cosine functions in the general solution because the exponential terms with imaginary numbers imply rotating vectors in the complex plane.

These equations are a specialized form of differential equations where the order is two, and they include no external input functions (non-homogeneous terms). They follow the general form \( ay'' + by' + cy = 0 \).
Such equations allow the use of characteristic equations to find solutions. Solutions are expressed as linear combinations of exponential functions corresponding to the roots of the characteristic equation. If the roots are complex, it takes the form of trigonometric functions.

• Real roots lead to exponential decay or growth.
• Complex roots lead to oscillatory behavior.

In our solved example, with roots \( r = \pm i\sqrt{5} \), the solution was an oscillatory one: \( y(t) = c_1 \cos(\sqrt{5} t) + c_2 \sin(\sqrt{5} t) \), reflecting the timeless, wave-like nature of solutions to such homogenous equations.