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In the realm of differential equations, particularly the Cauchy-Euler type, the characteristic equation serves as a key stepping stone. This equation is derived directly from the differential equation itself. By replacing derivatives in the original Cauchy-Euler equation with powers of the variable, one can arrive at a polynomial equation. For example, from the equation \(t^{2} y'' - 3t y' + 6y = 0\), we form the characteristic equation by substituting terms to find \(m^{2} - 3m + 6 = 0\).
This characteristic equation, \( am^{2} + bm + c = 0\), is quadratic in nature. Solving this quadratic equation gives the roots necessary for constructing the general solution. The nature of the roots (whether real or complex) significantly influences the form of the solution. Thus, identifying and solving the characteristic equation is a foundation step in analyzing Cauchy-Euler differential equations.

Once the characteristic equation is formulated, solving for its roots is the next critical task. When the discriminant \(b^2 - 4ac\) of the characteristic quadratic equation is less than zero, the roots are complex. In the given example, the discriminant turns out to be \(-15\).
This scenario results in complex roots of the form \(m = η ± iθ\), where \(η\) and \(θ\) are real numbers. For the example, we find roots \(3/2 ± i√15/2\). The presence of complex roots suggests oscillatory behavior in the solutions, often taking on a trigonometric form in the final expression. It's crucial to calculate the real and imaginary parts of these roots accurately, as they determine the shape of the general solution.

The culmination of identifying the characteristic equation and complex roots leads us to the general solution of the Cauchy-Euler differential equation. For equations with complex roots, the general solution incorporates exponential growth or decay, coupled with oscillatory components.
In our example, with roots \(m=3/2 ± i√15/2\), the general solution is expressed as:
\[ y = t^{3/2} [c_1 \cos(\frac{√15}{2} \ln|t|) + c_2 \sin(\frac{√15}{2} \ln|t|)] \]
This solution reflects the characteristics of both the exponential term \(t^{η}\) and the sinusoidal terms involving \(\cos\) and \(\sin\). Here, \(c_1\) and \(c_2\) are constants determined by initial conditions, if provided. This expression showcases the blend of exponential scaling and harmonic oscillation typical in Cauchy-Euler equations with complex roots.