Question

Without actually solving the differential equation (1-2sin x) y' + xy = 0,find a lower bound for the radius of convergence of power series solutions about the ordinary point x = 0 .

Short answer

Therefore, the solution is

$R=\frac{\mathrm{\pi }}{6}$

Step-by-step solution

Given Information

The given value is

(1-2 sin X ) y' + xy = 0.

Use Differentiate

Putting the given differential equation in standard form

$y"+\frac{x}{1-2\sin X}=0$

Identifying P(x) and Q (x).

$$\begin{gathered} P\left(x\right)=\frac{x}{1-2\sin X} \\ Q\left(x\right)=0 \end{gathered}$$$\frac{x}{1-2\sin X}=0$

Equating the denominator by zero

$$\begin{gathered} 1-2\sin X=0 \\ \sin x=\frac{1}{2} \end{gathered}$$

$X_1=\frac{\mathrm{\pi }}{6}$

Then, the given differential equation have a singular point at X₁

Finding as distance from 0 to the closest singular point. Hence $R=\frac{\mathrm{\pi }}{6}$