Question 2. Now we add the effect of gravity to the string. The force on the string now has an additional component straight down. Assuming that the deflection of the string is small (and ignoring some physical effects that we cannot legitimately ignore ${ }^{1}$ ), we find $$ \frac{\partial^{2}}{\partial t^{2}} utk \frac{\partial}{\partial t} u=c^{2} \frac{\partial^{2}}{\partial x^{2}} u-g $$ where $g$ captures the force of gravity. We assume the string still has clamped boundary conditions. $u(0)=u(L)=0$. (a) Assume that there is a steady solution to the equations $u(x, t)=U(x) $. Show that $U(x) $ solves $$ \frac{\partial^{2} U}{\partial x^{2}}=g / C^{2} $$ (b) Find $U(x)$ to satisfy the boundary condition. (c) Show that if $u(x, t) $ is a solution to equation (2), then $v(x, t)=(x, t)-u(x)$ solves equation (1). CS.VS. 1646||