I am trying to solve the partial differential equation $$ u_t - xtu_x = 0 $$ with the initial condition $$ u(x, 0) = \sin(x) $$. I have gone through the following steps and would like to confirm if they are correct.
From the partial differential equation $$ u_t - xtu_x = 0 $$, I obtained the characteristic equations: $$ \frac{dt}{1} = \frac{dx}{-xt} $$
Upon integration, I get: $$ \ln|x| = -\frac{t^2}{2} + c $$ where $c$ is another integration constant.
Since $ \frac{du}{0} $, I know that $ u $is constant along the characteristic curve. Therefore, the general solution can be represented as: $$ u(x, t) = f\left(e^{\frac{t^2}{2}}x\right) $$ where $ f $ is an arbitrary function.
Therefore, the particular solution is: $$ u(x, t) = \sin\left(e^{\frac{t^2}{2}}x\right) $$
Is my solution correct? If not, where did I go wrong and how can I correct it?
