In problems 1-5, find the Taylor series of the given functions about
$x_0=0.$ You may use your knowledge of the Taylor series of $e^x,$
$\sin(x),$ $\cos(x),$ and $\displaystyle \frac{1}{1-x}$ (the geometric
series).
$\displaystyle \frac{1}{1+x^2}$
$\displaystyle (x^2-1)\cos(x)$
$\displaystyle \cosh(x)$
$\displaystyle \sinh(x)$
$\displaystyle \exp(x)\sinh(x)$
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.
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
$$ x^2y'' - 2xy' + 2y = x^3\exp(x). $$
Find the general solution to
$$ x^2y'' + 6xy' + 6y = \cos(3x). $$
$$
\begin{align}
x^{a+ib} &= x^a x^{ib} \\
&= x^a \exp(ib\log|x|) \\
&= x^a \left(\cos(b\log|x|) + i\sin(b\log|x|) \right) \\
&= x^a\cos(\log|x^b|) + i x^a\sin(\log|x^b|)
\end{align}
$$
Recall from chapter 2 that if you have a complex
solution to a real homogeneous linear differential equation, then the real
and imaginary parts must both be solutions to the equation. These facts
may be helpful in the next several problems.
Find the general solution to
$$ x^2y'' + xy' + 4y = 0. $$
Find the general solution to
$$ x^2y'' + 3xy' + 10y = 0. $$
Find the general solution to
$$ x^2y'' - 3xy' + 8y = 0. $$
Find the general solution to
$$ (x^2+2x+1)y'' - (2x+2)y' + 2y = 0. $$
(Hint: make the substitution $u = x+1$).
Find the general solution to
$$ (x^2+4x+4)y'' + (7x+14)y' + 8y = 0. $$
(Hint: make the substitution $u = x+2$).
Find the general solution to
$$ (x^2+2x+1)y'' + (x+1)y' + y = 0. $$
(Hint: make the substitution $u = x+1$).
Find and classify (as regular or irregular) the singular points for
$$ (x^2+3x+2)^2y''+(x^2-4)y'+(2x-3)y=0. $$
Find and classify (as regular or irregular) the singular points for
$$ (x^4+x^3)y''+ x^2y' + (x+1)y=0. $$
Find and classify (as regular or irregular) the singular points for
$$ x^3 y'' + x^2y' + y = 0. $$
Find and classify (as regular or irregular) the singular points for
$$ (x^4 - 2x^2 + 1)y'' + (x^3+6x^2+5x)y' + (3x+2)y = 0. $$
Find and classify (as regular or irregular) the singular points for
$$ y'' + \frac{x}{x+1} y' + \frac{x+1}{x-1} y = 0. $$
Find and classify (as regular or irregular) the singular points for
$$ y'' + \frac{x^2+1}{x^3+x^2} y' + \frac{x^2-1}{x^2+1}y = 0. $$
Find and classify (as regular or irregular) the singular points for
$$ (\cos(x)-1)y'' + \sin(x)y' + \exp(x)y = 0. $$
Find and classify (as regular or irregular) the singular points for
$$ (\cosh(x)-1)y'' + \sinh(x)y' - xy = 0. $$
Where is the series solution about $x_0=0$ to the equation
$$x^2y''+(x^2-x)y'+(x-2)y=0$$
guaranteed to converge?
Where is the series solution about $x_0=-1$ to the equation
$$(x^2-2x-3)y''+(x^2+4x)y'+7y=0$$
guaranteed to converge?
Where is the series solution about $x_0=1$ to the equation
$$(x^4-2x^2+1)y'' + (x^2+3x+2)y'+x^2y=0$$
guaranteed to converge?
The hypergeometric equation with parameters $a$, $b$, and $c$
is $$z(1-z)\frac{d^2w}{dz^2}+[c-(a+b+1)z]\frac{dw}{dz}-abw=0.$$
Where are you guaranteed the series solution to the
hypergeometric equation about $z_0=0$ converges?
Find the roots of the indicial equation about $x_0=0$ for
$$ x^2y'' + (x^2+5x)y' + (x+3)y = 0. $$
Find the roots of the indicial equation about $x_0=0$ for
$$ 6x^2y'' + (2x^2+11x)y' + (2x^2+3x+4)y = 0 $$
Find the roots of the indicial equation about $x_0=0$ for
$$(x^3-x^2)y''+(x^2+x)y'+(2x-4)y=0$$
Find the roots of the indicial equation about $x_0=0$ for
$$ (x^4+6x^3+9x^2)y'' + (x^3-9x)y' + (33x+9)y = 0 $$
Find the roots of the indicial equation about $x_0=-3$ for
$$ (x^4+6x^3+9x^2)y'' + (x^3-9x)y' + (33x+9)y = 0 $$
Find the roots of the indicial equation about $x_0=0$ for
$$ (\cos(x)-1)y'' + \sin(x)y' + \exp(x)y = 0. $$
Spherical coordinates are defined by the transformations
Warning: physicists and mathematicians use different definitions of $\phi$
and $\theta$. This is a math class so these are the conventional
definitions used by mathematicians.
Write $\displaystyle \frac{\partial u}{\partial x}$ in terms of
$\rho,$ $\theta,$ and $\phi.$
Write $\displaystyle \frac{\partial u}{\partial y}$ in terms of
$\rho,$ $\theta,$ and $\phi.$
Write $\displaystyle \frac{\partial u}{\partial z}$ in terms of
$\rho,$ $\theta,$ and $\phi.$
Write $\displaystyle \frac{\partial^2 u}{\partial x^2}$ in terms of
$\rho,$ $\theta,$ and $\phi.$
Write $\displaystyle \frac{\partial^2 u}{\partial y^2}$ in terms of
$\rho,$ $\theta,$ and $\phi.$
Write $\displaystyle \frac{\partial^2 u}{\partial z^2}$ in terms of
$\rho,$ $\theta,$ and $\phi.$
If possible, solve the boundary value problem
$$ y'' + 5y' + 4y = 0, \qquad y(0)=1, \quad y(1) = 2. $$
If possible, solve the boundary value problem
$$ y'' + 4y' + 4y = 0, \qquad y(0)=0, \quad y(1) = 2. $$
If possible, solve the boundary value problem
$$ y'' + 2y' + 5y = 0, \qquad y(0)=0, \quad y(\pi) = 1. $$
If possible, solve the boundary value problem
$$ y'' + 2y' + 5y = 0, \qquad y(0)=1, \quad y(1) = 2. $$
For what values of $x_1$ are there infinitely many solutions to the
boundary value problem
$$ y'' + y = 0, \qquad y(0)=0, \quad y(x_1)=0. $$
For what values of $x_1$ are there infinitely many solutions to the
boundary value problem
$$ y'' + 4y'+8y = 0, \qquad y(0)=0, \quad y(x_1)=0. $$
For what values of $\alpha$ are there infinitely many solutions to the
boundary value problem
$$ y'' + \alpha y = 0, \qquad y(0)=0, \quad y(\pi) = 0. $$
For what values of $\alpha$ are there infinitely many solutions to the
boundary value problem
$$ y'' + \alpha y = 0, \qquad y(0)=0, \quad y(1) = 0. $$
For what values of $\alpha$ are there infinitely many solutions to the
boundary value problem
$$ y'' + 6y'+ \alpha y = 0, \qquad y(0)=0, \quad y(\pi) = 0. $$
For what values of $\alpha$ are there infinitely many solutions to the
boundary value problem
$$ y'' + 4y' + \alpha y = 0, \qquad y(0)=0, \quad y(1) = 0. $$
Make the substitution $u(t,x)=T(t)X(x)$ in the heat equation
$$ \frac{\partial u}{\partial t} = \kappa\frac{\partial^2 u}{\partial
x^2}$$
and show the equation separates into equations for $T$ and $X.$
Make the substitution $u(t,x)=T(t)X(x)$ in the wave equation
$$ \frac{\partial^2 u}{\partial t^2} = c\frac{\partial^2 u}{\partial
x^2}$$
and show the equation separates into equations for $T$ and $X.$
Show that for any values of $a$ and $b$, if the boundary value problem
$y''+ ay' + by=0$, $y(0)=0$, $y(1)=0$, has a non-zero solution, then it
has infinitely many solutions. Hint: You don't have to solve the
equation to show this. Just assume you have one non-zero solution and show
how to make infinitely many more solutions from that one.