Final answer:
The singular point of the given differential equation is x = 0, determined by the zero value of the coefficient of y′′ (x²). The regularity or irregularity of this point cannot be determined from the provided information.
Step-by-step explanation:
The question asks us to find all singular points of the given second-order differential equation and to determine whether each one is regular or irregular. The differential equation in question is:
x²y′′+2(ex−1)y′+(e−x cosx)y=0
To identify the singular points, we must look for values of x that make the highest order derivative coefficient (in this case, the coefficient of y′′) zero. Since the coefficient is x², the singular point is x = 0.
Next, we need to determine whether this point is regular or irregular. For that, we usually convert the equation into the standard form of a linear differential equation and examine the behavior of the coefficients as x approaches the singular point.
However, without a defined method for this particular equation, we cannot determine the regularity of the singular point directly from the given information.