Question

conclusion root 7.6 with proof​

Answer 1

Answer:

Step-by-step explanation:

The Method of Frobenius I

In this section we begin to study series solutions of a homogeneous linear second order differential equation with a regular singular point at x0=0, so it can be written as

x2A(x)y′′+xB(x)y′+C(x)y=0,(1)

where A, B, C are polynomials and A(0)≠0.

We’ll see that (eq:7.5.1) always has at least one solution of the form

y=xr∑n=0∞anxn

where a0≠0 and r is a suitably chosen number. The method we will use to find solutions of this form and other forms that we’ll encounter in the next two sections is called the method of Frobenius, and we’ll call them Frobenius solutions.

It can be shown that the power series ∑∞n=0anxn in a Frobenius solution of (eq:7.5.1) converges on some open interval (−ρ,ρ), where 0<ρ≤∞. However, since xr may be complex for negative x or undefined if x=0, we’ll consider solutions defined for positive values of x. Easy modifications of our results yield solutions defined for negative values of x.