Radius of convergence question

[josh146]

Homework Statement

What feature of the ODE explains your value for the radius of convergence of the series y2?
y2 is a series which satisfies the ODE and I found that it converges for $$\abs{2x^2} < 1$$.

Homework Equations

$$y2=x-\frac{2}{3}x^2-\frac{4}{15}x^5+ \cdots$$

ODE: $$(1-2x^2)\frac{d^2y}{dx^2} +4y = 0$$

The Attempt at a Solution

I presume this has something to do with the $$(1-2x^2)$$ in the ODE but I'm not sure. Can someone help?

 

[mighty2000]

Thank you for your question. The feature of the ODE that explains the value for the radius of convergence of the series y2 is the coefficient of the x^2 term, which is -2. In general, the radius of convergence of a power series solution to an ODE is determined by the coefficient of the highest order derivative term in the ODE.

In this case, the coefficient of the x^2 term is -2, which means that the radius of convergence of the series y2 will be 1/2, as you have correctly found. This is because the series will only converge for values of x that are less than the reciprocal of the coefficient, which in this case is 1/(-2) = -1/2. Therefore, the series will converge for values of x that are between -1/2 and 1/2, giving a radius of convergence of 1/2.

I hope this helps clarify the connection between the ODE and the radius of convergence of the series y2. Keep up the good work in your studies!