Question

Both power series solutions of centered at the ordinary point \(x = 0\) are guaranteed to converge for all \(x\) in which one of the following intervals?

(a) \(( - \infty ,\infty )\)

(b) \(( - 1,\infty )\)

(c) \(\left( { - \frac{1}{2},\frac{1}{2}} \right)\)

(d) \(( - 1,1)\)

Short answer

Therefore, \(\left( { - \frac{1}{2},\frac{1}{2}} \right)\) is the interval we were looking for.

Step-by-step solution

To Find the singular point

We were given the next DE Because of\(ln(x + 1),x = - 1\)is a singular point.

Notice that this is the nearest singular point to \(x = 0\)(which is ordinary point). That excludes intervals \(( - \infty ,\infty )\) and \(( - 1,1)\)as a solution . \(( - 1,\infty )\)is not the solution because the interval of convergence is centered at the center of the series.

Final Answer

Therefore, \(\left( { - \frac{1}{2},\frac{1}{2}} \right)\) is the interval we were looking for.