Determine the type of the following equation and reduce the PDE to its canonical form $u_{xx} + 4u_{xy} + 4u_{yy} + u = 0$.
We consider pdes in the form $$a_{11}(x,y)u_{xx}+2a_{12}(x,y)u_{xy}+a_{22}(x,y)u_{yy} +F(x,y,u,u_x,u_y)=0$$
Since in our case $a_{12}^2-a_{11}a_{22}=0$, we have that it is parabolic.
Then I think we find $$\frac{dy}{dx}=\frac{a_{12} \pm \sqrt{a_{12}^2-a_{11}a_{22}}}{a_{11}}=2$$ But then what?