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??


The general solution to the wave equation is the sum of the homogeneous solution plus any particular solution. The homogeneous solution is the solution to the equation when the RHS is equal to zero (with all the derivatives placed on the LHS, as in your very first equation). A particular solution is any solution that satisfies the equation with any non-derivative term (called inhomogeneous terms) placed on the RHS (the -g in your example). Particular solutions need not be unique. The homogeneous solution usually contains an infinite set, generally with undetermined constant coefficients. A particular solution is usually the result of a guess, in which experience helps. The sum of both homogeneous and particular solutions must satisfy the initial and boundary conditions, and this step usually evaluates the constants in the homogeneous solution. In your examples above, the first three terms on both RH sides is the homogeneous solution, and they are the same in both examples, since it's the same homogeneous equation for both. The last three terms on both RH sides are particular solutions, and they differ, since the inhomogeneous terms for both examples probably differ.


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