 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
 \[
x^3;
\]  \[
2x^2+y^2;
\]  \[
y^3y.
\]
 \[
 Let \(u\) be a harmonic function in
Archive  math 285 updates
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november 21

Find sin transform of the function on \([0,1]\)
given by \(1\);
 \(\cos(\pi x)\);
 \(x\).

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 u(x,0)=\(1\);
 u(x,0)=\(\cos(\pi x)\);
 u(x,0)=\(x\).
 Solve the heat equation
\[
math285, week of november 17
Read the textbook, chapters 9.5 and 9.7.
Homework (due by Monday, 12.1):
 Find the cosine transform of \(\sin(x)\) on \([0,\pi]\).
 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)
math 285, week of november 10
Read the textbook, chapters 9.4 and 9.6.
Homework (due by Monday, 11.17):
 Find sine transform of the function (\T(x)\) on \([0,1]\) equal to \(x\) for \(0\leq x\leq 1/2\) and to \(1x\) for \(1/2\leq x \leq1\).
 Using
november 6
 \[
\cos(2x)\sin^2(x);
\]  \[
\cos^3(x);
\]  \[
f(x)=\left\{\begin{array}{rcl} 1& \mathrm{if}& \pi\leq x<0;\\
1 & \mathrm{if}& 0\leq x \leq \pi.\\
\end{array}\right.
\]
october 31
 \[
2x
\]  \[
2x1
\]  \[
1+x.
\]
 \[
a=\pi, b=\pi/3;
\]  \[
a=\pi/3, b=\pi/3;
math285, week of october 27
Read the textbook, chapters 9.19.3.
Homework (due by Tuesday, 11.6):
 Find Fourier transform of the function on \((\pi,\pi)\) equal to \(0\) for \(\pi<x<0\) and to \(1\) for \(0\leq x<\pi\).
 Find cosine expansion of \(\sin(2x)\) on
october 24

Consider a mass \(m\) on a spring with spring constant \(k\) oscillating on a surface with the damping constant \(c=.01\).
 If \(m=1, k=16, c=.01\), find, approximately, how much shrinks the amplitude after each swing.
 Find the maximum amplitude of the
october 17


Solve using variation of parameters:
 \[
y”+3y’+2y=1/(e^x+1);
\]  \[
y”+y=1/\sin(x);
\]  \[
y”+4y=2\tan(x).
\]
 \[
 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
\[