# General Solution to the Wave Equation (Inhomogeneous)

I am having trouble finding the general solution to the wave equation for a plucked string subject to gravity. The question is:

Find the general solution to: $$u_{tt}-c^2u_{xx}=-g$$ Where $g$ is the magnitude of the acceleration due to gravity.

Now I have tried to factor it as usual, i.e.: $$\left(\frac{\partial}{\partial t}-c\frac{\partial}{\partial x}\right)\left(\frac{\partial}{\partial t}+c\frac{\partial}{\partial x}\right)\cdot u+g=0$$ But then what are the PDE's in this case? Are the Transport Equations as the following?: $$u_t+cu_x=-g, \>\>\>\> u_t-cu_x=-g$$ I know it's just one constant added to the equation but it's really confusing me. If anyone could help itd be appreciated!

Edit: I have come to the realization that $u$ can be: $$u(x,t) = f(x+ct)+g(x-ct)+\frac{gx^2}{2c^2}+bx+c, \>\>\>\> b,c\in\mathbb{R}$$ But however, we can also have the solution: $$u(x,t) = f(x+ct)+g(x-ct)-\frac{gt^2}{2}+bt+c, \>\>\>\>\>\> b, c\in\mathbb R$$ Or even a combination of the two: $$u(x,t) = f(x+ct)+g(x-ct)+\frac{gx^2}{4c^2}+bx+c-\frac{gt^2}{4}+\beta t+\gamma, \>\>\>\> b,c,\beta,\gamma\in\mathbb R$$ But my question only asks for $\boldsymbol{one}$ solution. Which is the correct one??