# Archive | courses

## math285, week of november 17

Read the textbook, chapters 9.5 and 9.7.

Homework (due by Monday, 12.1):

1. Find the cosine transform of $$\sin(x)$$ on $$[0,\pi]$$.
2. Using separation of variables, solve the heat equation on the interval $$[0,\pi]$$:
$u_{t}=u_{xx}; \quad u_x(0,t)=u_x(\pi,t)=0; u(x,0)=\sin(x) ## november 14 1. Find the solutions of the wave equation at time $$t=1/6,1/3,1/2$$ \[ u_{tt}=u_{xx}, u(0,t)=u(1,t)=0,$
on $$[0,1]$$
with initial data $$u_t(x,0)=0$$ and $$u(x,0)$$ is as shown:

2. Same question, but the initial data are
$$u(x,0)=0$$ and $$u_t(x,0)$$ is as shown:

3. Sketch the

## math 285, week of november 10

Read the textbook, chapters 9.4 and 9.6.

Videos to watch.

Homework (due by Monday, 11.17):

1. Find sine transform of the function (\T(x)\) on $$[0,1]$$ equal to $$x$$ for $$0\leq x\leq 1/2$$ and to $$1-x$$ for $$1/2\leq x \leq1$$.
2. Using

## november 6

• Find Fourier series for $$2\pi$$-periodic function defined on $$-\pi,\pi$$ by
1. $\cos(2x)\sin^2(x);$
2. $\cos^3(x);$
3. $f(x)=\left\{\begin{array}{rcl} 1& \mathrm{if}& -\pi\leq x<0;\\ -1 & \mathrm{if}& 0\leq x \leq \pi.\\ \end{array}\right.$
• Find the Fourier series for the function which is
• ## notes for lectures 10-14

here

Many thanks to James!…

here

## ece515, homework 4

$$\def\Real{\mathbb{R}} \def\Comp{\mathbb{C}} \def\Rat{\mathbb{Q}} \def\Field{\mathbb{F}} \def\Fun{\mathbf{Fun}} \def\e{\mathbf{e}} \def\f{\mathbf{f}}$$
Due by midnight of november 8.

1. Consider the system $$\dot{x}=Ax+bu$$,
$A=\left( \begin{array}{ccc} 1&1&0\\ 1&1&1\\ 0&1&1\\ \end{array}\right); b=\left( \begin{array}{c} b_1\\ b_2\\ b_3\\ \end{array}\right).$
• Derive conditions on $$b$$ making the system controllable.

## october 31

• Find Fourier series for
1. $2-|x|$
2. $2|x|-1$
3. $1+|x|.$
• Find the Fourier series for the function which is 1 on the interval $$[a,b]$$, and 0 elsewhere on $$[-\pi,\pi]$$, if
1. $a=-\pi, b=-\pi/3;$
2. $a=-\pi/3, b=\pi/3; • ## october 24 1. Consider a mass $$m$$ on a spring with spring constant $$k$$ oscillating on a surface with the damping constant $$c=.01$$. 1. If $$m=1, k=16, c=.01$$, find, approximately, how much shrinks the amplitude after each swing. 2. Find the maximum amplitude of the ## october 17 1. Solve using variation of parameters: 2. \[ y”+3y’+2y=1/(e^x+1);$
3. $y”+y=1/\sin(x);$
4. $y”+4y=2\tan(x).$
1. One of the solutions of the linear differential equation of the second order
$y”+a_1(t)y’+a_2(t)y=0$
is given; find another one using the Liouville formula
\[