Vtyjkyuk

I came across a partial differential equation (IVP PDE) that I would like to solve:

$$ u(x,0)=0$$

This should be a quasilinear PDE, and is in the format of a Cauchy Problem, in the form of:

$$au_x+bu_y=c$$
Such that a, b, and c are constants.

In my particular case, I have:

$$
left{
beginarrayc
a=cos(ky) \
b=1 \
c=ax^2
endarray
right.
$$
Using the Lagrange-Carpit Equations: $$fracdxcos(ky)=fracdy1=fracduax^2$$
Rearranging gives: $$fracdxdy=cos(ky)$$
$$dy=fracduax^2$$
I then canclulated $dx-dy$:
$$du=(ax^2cos(ky)+1)dy-dx$$
Integrate to obtain $u(x,y,a,k)$:
$$u(x,y,a,k)=frac1kax^2sin(ky)+y-x+C_1$$
I applied the initial condition of $u(x,0)=0$ and made $x=0$, which gave $C_1=0$.
The question is thus finalised.

This however does not look correct. Am I doing the right thing?

I would also like to see what the characteristic curves look like, what are these? Are they essentially the terms $a, b, c$ I obtained above? Or rather, I simply differentiate those three variables in terms of $s$. Essentially calculating $fracdyds$, $fracdxds$ and $fracdzds$.