2. Determine solution of the following partial differential equation a²U/əx² = a² au/at subjected to: U (x, 0) = 150 = U(o, t) = 0 U (1,t) = 0

6. Reduce the ODE y"" + 2(y')2 - exp(-x)y = 0 to a fundamental system of ODES.

7. Consider the function f(x, y) = cos (y/x) 1. Is this a homogeneous function? Why? 2. If yes, what is the degree of this functions?

If z = y³ – xy² - 2x³, find the partial derivatives ∂z/∂x and ∂z/∂y Remember not to use any decimal points in your answer.

4. Compute the work done by the vector field F = (r²e² + 2y³)i + (y³ − y² + 2x³ + 2x) j along closed curve C = C₁UC₂ UC3 UC4 as shown.

1 Using equation (5.3b), find the Taylor series around x = -pi/2 of the function f(x) = sin(2x), up to and including the term in h². Remember not to use any decimal points in your answer. f(xo + h) ~

Consider the function f(x, y) = In (√√Ax² + 6y²). a. Find fxx. fxx (x, y) =

4 If w = u² cos(v) where u(t) = 3 e^t and v(t) = 3 t³, find ·

a. Find ztt for z(x, y, t) = cos(√9+16t) sin(3x) sin(4y). b. Does u = sin(√25 t) sin(3x) sin(4y) satisfy the membrane equation utt = uxx + uyy ?

A violin string produces a vibration where we have that A > 0, that a and b are arbitrary constants, and that utt = c²uxx for some c. Find E>0 in terms of A. E=