Final answer:
The question asks for a particular solution to a differential equation using complex-valued trial functions, typically solved by assuming a solution form and determining constants.
Step-by-step explanation:
The question relates to finding a particular solution to a differential equation using a complex-valued trial solution. When faced with a differential equation of the form sin y'' +2y' +2y=2e⁻¹ sinx, adopting a systematic approach with suitable trial functions is crucial. Commonly, the method of undetermined coefficients or the variation of parameters is applied to solve such non-homogeneous linear differential equations. However, the exact steps to find the solution involve assuming a trial solution that will satisfy the given equation and then determining the constants by substituting it back into the equation.