91Ó°ÊÓ

Find the radius of convergence of each of the series in Exercises 1-12. $$ \sum_{n=0}^{\infty} \frac{x^{n}}{n+1} $$

In Exercises \(1-4\), the indicial equation corresponding to the given differential equation has equal roots. Find a fundamental set of solutions for the given differential equation. $$ x^{2} y^{\prime \prime}+3 x y^{\prime}+(1-2 x) y=0 $$

Differential equations do not usually have power series solutions near singular points, even if they are regular. As an example, consider the equation $$ x^{2} y^{\prime \prime}+4 x y^{\prime}+2 y=0 . $$ which is an Euler's equation, and has a regular singular point at \(x_{0}=0\). Show that if the series \(y(x)=\sum_{n=0}^{\infty} a_{n} x^{n}\) is a solution, then \(a_{n}=0\) for all \(n\).

In Exercises 1-9, classify each singular point of the given equation. $$ x^{2} y^{\prime \prime}+4 x y^{\prime}-2 x y=0 $$

Find the radius of convergence of each of the series in Exercises 1-12. $$ \sum_{1=0}^{\infty} \frac{x^{n}}{\sqrt{n+3}} $$

In Exercises 1-9, classify each singular point of the given equation. $$ x^{2}(1-x) y^{\prime \prime}+(x-2) y^{\prime}-3 x y=0 $$

In Exercises 1-14, solve the given equation exactly using a technique from a previous chapter. Then find a power series solution and verify that it is the series expansion of the exact solution. $$ y^{\prime}+2 y=0 $$

In Exercises \(1-4\), the indicial equation corresponding to the given differential equation has equal roots. Find a fundamental set of solutions for the given differential equation. $$ x^{2} y^{\prime \prime}-x(1+x) y^{\prime}+y=0 $$

There are many identities relating the Bessel functions of different orders and their derivatives. In Exercises 2-7, you are asked first to derive them, and then to use them. By differentiating the series term by term, prove that $$ \left[x^{-p} J_{p}\right]^{\prime}=-x^{-p} J_{p+1} . $$

In Exercises \(1-4\), the indicial equation corresponding to the given differential equation has equal roots. Find a fundamental set of solutions for the given differential equation. $$ x y^{\prime \prime}+(1-x) y^{\prime}-y=0 $$