# The number of characteristic curves of the PDE

The number of characteristic curves of the PDE $(x^2+2y)u_{xx}+(y^3-y+x)u_{yy}+x^2(y-1)u_{xy}+3u_x+u=0$ passing through the point $x =1$, $y =1$ is
1. $0$
2. $1$
3. $2$
4. $3$

Given a second order linear PDE of the type you ask about, written in the form $$au_{xx}+2bu_{xy}+cu_{yy}+du_x+eu_y+fu=g,$$ the characteristics are the (real) solutions of the ODE $$\frac{dy}{dx}=\frac{1}{a}(-b\pm\sqrt{b^2-ac}).$$ By identifying the values of $a,b,c$ at $(x,y)=(1,1)$, can you see how many real solutions there are in your case?