1. find u(r,theta) when r is equivalent to sqrt(x(1)^2+x(2)^2) less than or equal to 2

laplacian of u equals zero, in r less than or equal to 2

u(2,theta)=cos(3*theta) when 0 <theta<2pi

2.find all u(x)=u(x1,x2) in 0<x1<2; -1<x2<1 when laplacian of u equals zero

BCs are du/dx1(0,x2)=du/dx2(2,x2)=0 for previous constraints on x1, x2

du/dx2(x1,1)=1+cos(3*(x2))

du/dx2(x1,-1)=1+cos(3*(x2))

3.let u(x,t); x in R^1, t>0 satisfy

u_tt - u_xx +2u_x +3u = 0; u(x,0)=f(x); u_t(x,0)=g(x)

Let uhat(xi,t)=1/sqrt(2pi) * integral from -inf to +inf (exp[-i*xi*x]*u(x,t)dx) be the fourier transform of u wrt x

i)find uhat(xi,t) in terms of fhat(xi) and ghat(xi)

ii)Give an integral for u(x,t) in terms of fhat(xi) and ghat(xi), don't evaluate integral

4. find u(x,t)=u(x1,x2,x3,t) for x in R^3, t>0 if

u_tt - laplacian u = 0; u(x,0) = 0; u_t(x,0)=aX_2(x) which is equivalent to a if |x|<2 and 0 if |x|>2.

i)give an explicit formula(s) in simplest terms for u. Evaluate any integrals.

ii)give all values of t for which u(0,0,5,t) does not equal 0

5. find the solution u(x,t), x in R^3, t>0 to

u_tt - c^2 * laplacian u = del(x1)del(x2)del(x3)del(t)

u(x,0)=0

u_t(x,0)=0

Evaluate all integrals to get the fundamental solution of this equation.

Again sorry if i'm not supposed to post more than one problem at a time, any help would be greatly appreciated.