The two-dimensional wave equation describing the vibrations of an infi- nite string is Θη Ot2 2021 მ2 where n = n(x,t), — -x< x < ∞ and c > 0 is a constant. Its general solution can be written as n(x,t) = F(x-ct) + G(x + ct), with F and G arbitrary smooth functions. If at time t = 0 the shape of the string is n(x, t = 0) at 9 = 5+x4 and the string is released with velocity (x, t = 0) = 0, determine n(x,t) for t > 0.

A system is described by the action S[r, r'] = [ dt (x² + 4rª + mr³4), r(t) where m is a constant. Find the Euler-Lagrange equation for r = which extremises S. Interpret each of the terms that appear in the Euler- Lagrange equations in terms of Newton's second law. Find a conserved quantity for this system.

Let f be the function defined as, Ր f(x) = [* sin (t³) dt. Determine an expression for f(x) as a power series.

Let X be a random variable such that E[X] = μ, V[X] =σ², max[X] = My min(X) = m with 0 Not egual to m < M and consider its normalization: E[Y] =μ M-m Determine whether the following statements are true or false. Perform the corresponding calculations. E[Y] μ V[Y] = M-m 2 σ M-m