au Given uit, x), satisfying 34+ + = (24) = 0. Assume that u E CT([0, 0) x R") is a solution of (3.34). By analogy with Theo- rem 3.7, a characteristic of (3.34) is defined as a solution of X (3.0) dx dt B 16 0) = 12 algus(t)=(t, (t)=(t, x(t)), x(0)=xo. (3.35) fan od ot lullsise vd (a) Assuming that x(t) solves (3.35), use the chain rule to compute d2x/dt2. U (b) Differentiate (3.34) with respect to x and then restrict the result to (t, x(t)), where x(t) solves (3.35). Conclude from (a) that to golauzoq emas yol bod fat dx yd navig ai (0 = 0. dt2 Hence, for some constant vo (which depends on the characteristic), x(t)=xo+vot. 22 to 2isbom ai 216sqm ledi nouses amiliesuo sigmic & e 101 mol & bai 19 2352 (c) Show that the Lagrangian derivative of u along x(t) satisfies 101 stormot s ball of 1002 mod Du Dt = 1 .2 2 Votos to bortom si sal) (a) noitibes itini di nuvis (3) noilor st implying that 1 u(t, xo+vot) = u(0, xo) + (d) Use this approach to find the solution u(t, x) under the initial condition u(0, x) = x. aqu2 (d) (For the characteristic starting at (0, xo), note that you can compute vo by eval- uating (3.35) at t=0.) 0