Burgers’ equation  $u_t + u u_x = −x $

Let the characteristic curves $\Gamma$ be parametrized by $s$ as follows $(s,0,f(s))$. We have \[\frac{dx}{dt}=z, \frac{dy}{dt}=1, \frac{du}{dt}=-x.\] Hence $y=t$, and $x$ and $u$ satisfy the coupled system \[\frac{dx}{dt}=u, \frac{du}{dt}=-x\] which can be written as a second order ODE \[x'' + x = 0, x(s, 0) = s, x'(s, 0) = u(0) = f(s).\] This is a second order linear ODE with constant coefficients whose solutions are given by \[ x(s, t) = s \cos t + f(s) \sin t,\] and hence \[u(s, t)= x'(t) = ?s \sin t + f(s) \cos t.\] Substituting $t=y$ we get \[ x= s \cos y + f(s) \sin y,\] \[(*)    u= ?s \sin y + f(s) \cos y.\] Hence \[ x \cos y= s \cos^2 y + f(s) \sin y \cos y,\] \[-u\sin y= s \sin^2 y - f(s) \sin y \cos y.\] Adding the two equations together we get \[x \cos y ? u \sin y = s(\cos ^2 y + \sin ^2 y) = s.\] Substituting $s$ in (*) we get \[u(x, y) = f(x \cos y ? u \sin y) \cos y ? (x \cos y ? u \sin y) \sin y.\]