Question

Find a minimum value for the radius of convergence of a power series solution about Xo- (x2 - 13x + 42)y” - 4xy - y = 0; x = 0 Select the correct answer below and, if necessary, fill in the corresponding answer box to complete your choice. (Type an exact answer.) O A. The radius of convergence is finite and at least OB. The radius of convergence is infinite.

Answer 1

Solution:-

Given that

$(x^2-13x+42)y''-4xy'-y=0$

$x_0=0$ is an ordinary point .

A power series solution conveyes atleast on some interval defined by

- col <R

where

R is the distance from $x_0$ to the closest singularity or closest singular point.

$y''-\frac{4x}{(x-6)(x-7)}y'-\frac{1}{(x-6)(x-7)}y=0$

Therefore

x = 6 & x = 7 are 2 singular points of differential equation.

a)

Radius of convergence is finite & atleast 6

and option (b) is incorrect for a power series solution about $x_0=0$ ,

but $y(x)=0$ is also a solution.

& radius of convergence is $\infty$ .

but you are asking for a power seires solution about $x_0=0$

So Option a is correct

b is incorrect

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