# 12-267/Homework Assignment 7

This assignment is due in class on Friday November 23. Here and everywhere, neatness counts!! You may be brilliant and you may mean just the right things, but if your readers have a hard time deciphering your work they will give up and assume it is wrong.

1. $\sum_{n=0}^\infty 2^nx^n$.
2. $\sum_{n=0}^\infty\frac{n}{2^n}x^n$.
3. $\sum_{n=0}^\infty\frac{(2x+1)^n}{n^2}$ near $x=-\frac12$.

Task 2. Solve the equation $(y')^2=1-y^2$ with $y(0)=0$ and $y'(0)>0$ using power series up to and including the coefficient of $x^5$. Then compare your result with the Taylor expansion of the exact solution.

Task 3. Find the recurrence relation defining the power series solutions of the following equations:

1. $y''-xy'-y=0$ with $y(0)=1$ and $y'(0)=0$.
2. $y''-xy'-y=0$ with $y(1)=1$ and $y'(1)=0$.
3. $(1-x)y''+y=0$ with $y(0)=0$ and $y'(0)=1$.

Task 4. In view of the theorem about convergence of power series solutions (Fuchs' theorem), give a lower bound on the radius of convergence of the series solution of the equation $(x^2-2x-3)y''+xy'+4y=0$ near $x=0$.

1. Find a recurrence relation satisfied by $a_n:=\begin{pmatrix}2n\\n\end{pmatrix}$.
2. Find a differential equation satisfied by $y(x):=\sum_{n=0}^\infty\begin{pmatrix}2n\\n\end{pmatrix}x^n$.
3. Solve that equation to determine $y(x)$ in general, and $\sum_{n=0}^\infty\frac{1}{5^n}\begin{pmatrix}2n\\n\end{pmatrix}$ in particular.