determine a Frobenius series solution to the given differential equation and use the reduction of order technique to find a second linearly independent solution on (0,∞). Q. xy′′−xy′+y =0.

Consider the following differential equation: (x² + 1)y″+8xy′+7y = 0. Suppose that this equation has a series solution of the form: y = Σ (x Σαn (2 - 7)" n=0 on some open interval that contains the point x0 = 7. Determine the singular points z in the complex plane. Enter the solutions as a comma-separated list. z = Making an appropriate sketch that shows the geometry of the problem, find the minimum radius of convergence p as the distance from the expansion point to the nearest singular point in the complex plane. Note: if the answer is a radical that can be reduced, make sure to simplify your answer. P= Find the minimum interval of convergence, I = (xop, xo+p). I =

Solve the following differential equation and express the solution of this differential equation in terms of an integral. (Hint: This is a linear differential equation with a forcing term and can be treated as a linear differential equation with nonconstant coefficients.) dx/dt = a*x + f(x) where f is a function of x and a is a constant.

2.