Differential Equations Answers

Solve the Cauchy problem on a semi-infinite domain:
u_tt - 4u_xx = 0 on 0<x<infinity, t>0
u(x,0) = xe^(-x)
u_t(x,0) = 0
u(0,t) = 0

Q1 : Differentiate e^2x In(x^2 - 3) with respect to x.

Q2: Differentiate (3x^2 + 2x) e^-x with respect to x.

using chain rule find dx/dy of y= u/5+7 and u=5x-35?

Hi! I have got 35 coordinates (x;y) which seem to form a function of the second degree. I want to know the basic expression for this function, even if I just have the coordinates. These are:
1,3 3,15 4,26 5,37 6,50 7,67 8,74 9,85 10,102 11,125 12,138 13,155 14,174 15,197 16,228 17,257 18,280 19,309 20,352 21,383 22,420 23,463 24,522 25,563 26,610 27,653 28,700 29,761 30,832 31,911 32,970 33,1037 34,1098 35,1181

Thanks in advance!
Gunnar A.

differential of y=(1+cos^2(3x))^2

I got (1+cos^2(3x))(cos3x)(-sin3x)
but my answer key says (-3cos3xsin3x)/(root of (1+cos^2(3x)))

evalaet the expression or x=6
x2-15
-x-9

Solve the Cauchy problem equation:

u_x + u_y = u^2, u(x,0) = 1.

Consider the equation yu_x - xu_y = 0, (y>0). Check for each of the following initial conditions whether the problem is solvable. If it is solvable, find a solution. If it is not, explain why:

a) u(x,0) = x^2
b) u(x,0) = x
c) u(x,0) = x, x>0

Solve the equation xu_x + (x+y)u_y = 1 with the initial condition u(1,y) = y. Is the solution defined everywhere?

Prove that the system:

U_x = 2.999999x^2y + y
U_y = x^3 + x

has no solution at all.