# Archive | math285

## december 5

1. Solve the Dirichlet problem in the disk $$D=\{x^2+y^2\leq 4\}$$ (i.e. find a harmonic function $$u$$ such that on the boundary $$u$$ is equal to
1. $x^3;$
2. $2x^2+y^2;$
3. $y^3-y.$
2. Let $$u$$ be a harmonic function in

## november 21

1. Find sin transform of the function on $$[0,1]$$
given by
1. $$1$$;
2. $$\cos(\pi x)$$;
3. $$x$$.
2. Solve the heat equation
$u_t=\frac{1}{2} u_{xx},$
on $$[0,1]$$ with boundary conditions $$u(0,t)=u(1,t)=0$$ and initial values
1. u(x,0)=$$1$$;
2. u(x,0)=$$\cos(\pi x)$$;
3. u(x,0)=$$x$$.
3. Solve the heat equation
$## 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
• ## 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; • ## math285, week of october 27 Read the textbook, chapters 9.1-9.3. Homework (due by Tuesday, 11.6): 1. Find Fourier transform of the function on $$(-\pi,\pi)$$ equal to $$0$$ for $$-\pi<x<0$$ and to $$1$$ for $$0\leq x<\pi$$. 2. Find cosine expansion of $$\sin(2x)$$ on ## 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
\[