Elementary Differential Equations

Series Solutions

$\displaystyle \frac{1}{1+x^2}$
$\displaystyle (x^2-1)\cos(x)$
$\displaystyle \cosh(x)$
$\displaystyle \sinh(x)$
$\displaystyle \exp(x)\sinh(x)$
If $\displaystyle y(x)=\sum_{n=0}^{\infty}a_nx^n$, then $y(0)=a_0$ and $y'(0)=a_1$. Show this pattern does not continue. In particular show $y''(0)\ne a_2$.

Suppose $\displaystyle y(x)=\sum_{n=0}^{\infty}a_nx^n$ where the sum for $y(x)$ has only even terms, i.e. $a_{2n+1}=0$ for all $n$. Show that $y(x)=y(-x)$, i.e. $y$ is an even function.

Suppose $\displaystyle y(x)=\sum_{n=0}^{\infty}a_nx^n$ where the sum for $y(x)$ has only odd terms, i.e. $a_{2n}=0$ for all $n$. Show that $y(x)=-y(-x)$, i.e. $y$ is an odd function.

Show that the solution to $$ \begin{align} y''+2xy'+(x^2+1)y=0, \\ y(0)=1, y'(0)=0 \end{align} $$ is an even function.

Show that the solution to $$ \begin{align} y''+2xy'+(x^2+1)y=0, \\ y(0)=0, y'(0)=1 \end{align} $$ is an odd function.

Show that the solution to $$ \begin{align} y''+2xy'+(x^2+1)y=0, \\ y(0)=1, y'(0)=1 \end{align} $$ is neither an even function nor an odd function.

Defining a function in terms of a series written out in the form $\displaystyle 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} + \cdots$ is generally less useful than a compact formula such as $\displaystyle \sum_{n=0}^{\infty}\frac{x^n}{n!}$ which gives a formula for every term. Solve the initial value problem $y''+xy'+y=0$, $y(0)=1$, $y'(0)=0$, and then find a pattern for the terms so you can write the solution in $\sum$ form.

Defining a function in terms of a series written out in the form $\displaystyle 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} + \cdots$ is generally less useful than a compact formula such as $\displaystyle \sum_{n=0}^{\infty}\frac{x^n}{n!}$ which gives a formula for every term. Solve the initial value problem $(1-x^2)y''+2y=0$, $y(0)=0$, $y'(0)=1$, and then find a pattern for the terms so you can write the solution in $\sum$ form.

The polynomials in the next four problems are called the Hermite polynomials. They are useful in certain situations involving the normal probability distribution.

Solve the initial value problem $$ y'' - 2xy' + 4y = 0, \qquad y(0)=1, \quad y'(0)=0.$$ Show that the series solution terminates so your solution is actually a polynomial.

Solve the initial value problem $$ y'' - 2xy' + 6y = 0, \qquad y(0)=0, \quad y'(0)=1.$$ Show that the series solution terminates so your solution is actually a polynomial.

Let $p$ be an even integer. Show that the solution to the initial value problem $$ y'' - 2xy' + 2py = 0, \qquad y(0)=1, \quad y'(0)=0 $$ is an even polynomial of degree $p$.

Let $p$ be an odd integer. Show that the solution to the initial value problem $$ y'' - 2xy' + 2py = 0, \qquad y(0)=0, \quad y'(0)=1 $$ is an odd polynomial of degree $p$.

We have claimed that the lower bound for the radius of convergence by the distance to the nearest singular point is usually the actual radius of convergence. In the next four problems you will justify this claim for the problem $$ \begin{align}(x^2+4)y''+y&=0 \\ y(0)=1,\quad y'(0)=&0 \end{align} $$

Find the singular points for the equation $(x^2+4)y''+y=0$ and use them to find the lower bound for the radius of convergence about $x_0=0.$

Find the recurrence relation for the series solution to $(x^2+4)y''+y=0$

Using the recurrence relation from the previous problem, show that the series solution $\displaystyle y(x)=\sum_{n=0}^{\infty}a_nx^n$ to the initial value problem $$ \begin{align} (x^2+4)y''+y&=0 \\ y(0)=1,\quad y'(0)=&0 \end{align} $$ has $a_n=0$ for all odd $n,$ so the ratio of successive terms is $\displaystyle \frac{a_{n+2}x^{n+2}}{a_nx^n}.$

By the ratio test, the series solution in the previous problems converges if $\displaystyle \lim_{n\to\infty} \left| \frac{a_{n+2}x^{n+2}}{a_nx^n} \right| < 1$ and diverges if $\displaystyle \lim_{n\to\infty} \left| \frac{a_{n+2}x^{n+2}}{a_nx^n}\right| >1.$ Use the recurrence relation and this test to show the series converges for $|x|<2$ and diverges for $|x|>2$, hence the radius of convergence is indeed 2.

Find the general solution to $$ x^2y'' + xy' - 4y = 0. $$

Find the general solution to $$ x^2y'' + 5xy' + 3y = 0. $$

Find the general solution to $$ x^2y'' - 5xy' + 8y = 0. $$

Find the general solution to $$ 2x^2y'' - 5xy' + 3y = 0. $$

Find the general solution to $$ 3x^2y'' - 2xy' + 2y = 0. $$

Find the general solution to $$ x^2y'' - 4xy' + 6y = x^4. $$

Find the general solution to $$ x^2y'' + 4xy' + 2y = \exp(2x). $$

Find the general solution to $$ x^2y'' - 2xy' + 2y = x^3\exp(x). $$