Final answer:
The general solution for the Euler equation with real and distinct roots α and β is the linear combination of the exponential functions of these roots, expressed as eαt + eβt. This excludes oscillatory components as the roots are not complex. Verification involves differentiation and substitution into the original equation.
Step-by-step explanation:
To determine the general solution for the given Euler equation with characteristic roots α (alpha) and β (beta) that are real and distinct, one would typically start by examining the form of the solution given by the characteristic equation. Since Euler's formula states that eiφ = cos(φ) + i sin(φ), this allows us to express solutions involving complex exponents in terms of real functions like sines and cosines when dealing with differential equations.
However, given that α and β are real and distinct, the solution will not include the oscillatory sine and cosine functions as it would for complex roots. Instead, the general solution for the Euler equation with real, distinct roots is a linear combination of the exponential functions of these roots. Therefore, the correct general solution is a) eαt + eβt, as each term in the solution corresponds to an individual root of the characteristic equation.
The step-by-step substitution that would lead to confirming this solution involves taking these forms as a potential solution to the Euler equation and then demonstrating that they satisfy the differential equation through differentiation and substitution. Finally, applying the initial or boundary conditions would yield the specific constants that customize the general solution to a particular problem.