Final answer:
To solve the Cauchy-Euler equation, a characteristic equation is formed from the general solution format y(t) = t^r, which helps find the roots for constructing the solution that fits the initial conditions y(1) = 0 and y'(1) = -6.
Step-by-step explanation:
The Cauchy-Euler equation t²y"(t) - 8ty'(t) + 20y(t) = 0 is a second-order linear homogeneous differential equation, where the general solution can often be found in the form of y(t) = t^r. Plugging this form into the equation, we get a characteristic equation whose roots will determine the form of the solution. To apply the initial conditions, y(1) = 0 and y'(1) = -6, we substitute t = 1 into the general solution and its derivative respectively.
After finding the roots of the characteristic equation, the general solution will often appear in a form that is a linear combination of the found solutions raised to the power of t. Since the initial conditions are given, we will have a system of equations that can be solved for the constants in the general solution. The final answer will include only the terms that satisfy the initial conditions.