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Done 5 of 5 पण DUIVO ult Uxx WWU morum 0W-vanu "Ow=1. 6.4.12 If u, = c(t,x)uz, the characteristic curves in the x- t plane satisfy dx/dt = -c(t,x). Along such a curve u is constant because du = u,dt + uzdx = u,dt + uz(-cdt) =(u, – cux)dt = 0. = (a) Find the solution X(t) of dX /dt = e' that has X = x at t = 0) (b) Verify that u= u(X(t)) solves u, = e'ux (c) Do the same for the ordinary differential equation dX/dt = X and the partial differential equation u, = xux. 6.4.13 Solve the wave equation 4, =c?Uxx starting at rest (vo =0) from a square wave: Up = 1 for lx| = 1, uo = 0 for [x] > 1. Graph the solution at t = 2. 6.4.14 How can sin x cost and cos x sint oscillate in place, when their sum sin(x + t) travels to the left? = = +