Solve PDE in 2D Problem How should I go about solving this PDE:
$$ \phi_x+\phi_y=x+y-3c  $$
Where $\phi = \phi(x,y)$, $c$ is a constant, and $\phi$ is specified on the circle
$$ x^2+y^2=1  $$
My Attempt to solve it I would like to use the method of characteristics, but then I get stuck because of the given initial condition. In fact, so far I have the characteristics equations
$$ \dot{{z}}(s)=x+y-3c $$
$$ \dot{{x}}(s)= 1 $$
$$ \dot{{y}}(s)= 1 $$
The last two are easy to solve but then I am not sure how to use the initial condition.
If you know of a different/easier method to solve this PDE, feel free to let me know, thanks!

As you say, the characteristic equations are $\dot{z} = x + y - 3 c$, $\dot{x} = 1$, $\dot{y} = 1$.  So the characteristic curves are $x = x_0 + s$, $y = y_0 + s$, i.e. $x - y = \text{constant}$.  But there's a problem with specifying the initial conditions on the circle $x^2 + y^2 = 1$: the characteristic curves through most points either don't intersect the circle at all (so the initial condition doesn't determine $\phi$ there) or intersect it in two points (so
the initial conditions might not be consistent).
