Construct a solution to the wave equation ô²u(x,t) _ ô²u(x,t) Ôx? over the range 0 < x < +o and 0 < t, given the one boundary condition u(0,t) = t/2, for 0 < t and the two initial conditions du(x,t) u(x,0) = x² and 1 for 0

Consider the wave equation: utt = uxx, −∞ < x < ∞, t > 0 with ut(x, 0) = 0 and u(x, 0) = ( 0, x < 0 and x > π) u(x, 0) = cos2 x, 0 ≤ x ≤ π a)Find a solution to this problem using the D’Alembert’s formula. Then, plot the solution at t = 0, π /4 , π /2 , π b)Find a solution to this problem using the Fourier transform and show that your solution is the same as part (a).

3) Can you answer just a,b,c

from t=1 to t=2. Hint: perfect square. ""(t)/(Ir'(t) x r''(t) 1)2. Compute a constant. Here x denotes cross da and b are constants. al/first order wave equation Utt=Uxx- f(x,y)=(x²/2)+3y³+9y²-3xy+9y-9x. -3) is Saddle Point (12.1) is local MIN ounded by the lines x=0, x=2, y=x is the region in the first quadrant at the origin. 7) Calculate SSS x²y²z² du where {AL F = {(x₁1₁2) | -1 ≤ x ≤ 1, -147, 51, -1 ≤ 2 ≤₁3