I was just introduced to method of characteristics for solving PDE's. We solved the wave equation that is inifinitely long using this method. However I am very confused about this method. Here is a question from the book I am trying to do. It says to solve the first order equations for $u(x,t)$ using this method. No initial conditions are given. $$ 2 \frac{\partial u}{\partial t} + 3 \frac{\partial u}{\partial x} = 0$$
There is an example in the book (Haberman) that solves PDE in this form $$ \frac{\partial u}{\partial t} + c \frac{\partial u}{\partial x} = 0$$
no naturally, for my solution I divide everything by $2$ to get $$ \frac{\partial u}{\partial t} + \frac{3}{2} \frac{\partial u}{\partial x} = 0$$ Then I let $u(x,t) = u(x(t),t)$ and so if I take the derivative of this I get $$\frac{du}{dt}(x(t),t) = \frac{\partial u}{\partial t} + \frac{\partial u}{\partial x} \frac{dx}{dt}$$ so it must mean that $\frac{dx}{dt} = \frac{3}{2}$ so this is easy enough so we get a solution $$ x = \frac{3}{2}t + x_0$$
okay but I need this in terms uf $u(x,t)$. I am lost at this point on what to do since the book continues using intial conditions.
Similarly I have a second question that is bothering me even more $$ 2 \frac{\partial u}{\partial t} + 3 \frac{\partial u}{\partial x} = 1$$