The problem describes a vibrating string of length (L) with fixed ends, zero initial velocity, and an initial deflection (f(x)). The solution can be obtained using separation of variables and the Fourier sine series. The wave equation for a string is: [ \frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2}, \quad 0 < x < L, ; t>0 ] with boundary conditions: [ u(0,t) = 0, \quad u(L,t) = 0 ] and initial conditions: [ u(x,0) = f(x), \quad u_t(x,0) = 0 ] --- ### Step 1: General solution For fixed ends and zero initial velocity, the solution is: [ u(x,t) = \sum_{n=1}^{\infty} B_n \sin\left(\frac{n \pi x}{L}\right) \cos\left(\frac{n \pi c t}{L}\right) ] where (B_n) are the Fourier sine coefficients of (f(x)): [ B_n = \frac{2}{L} \int_0^L f(x) \sin\left(\frac{n \pi x}{L}\right) dx ] --- ### Step 2: Compute Fourier coefficients The given piecewise function: [ f(x) = \begin{cases} Ax & 0 \le x \le L/4 [2mm] AL/4 & L/4 \le x \le 3L/4 [1mm] A(L-x) & 3L/4 \le x \le L \end{cases} ] Split the integral into three parts: [ B_n = \frac{2}{L} \left[ \int_0^{L/4} Ax \sin\frac{n\pi x}{L} dx + \int_{L/4}^{3L/4} \frac{AL}{4} \sin\frac{n\pi x}{L} dx + \int_{3L/4}^{L} A(L-x) \sin\frac{n\pi x}{L} dx \right] ] --- ### Step 2a: First integral [ \int_0^{L/4} Ax \sin\frac{n \pi x}{L} dx = A \int_0^{L/4} x \sin\frac{n \pi x}{L} dx ] Use formula (\int x \sin(ax) dx = -\frac{x}{a} \cos(ax) + \frac{1}{a^2} \sin(ax)): [ \int_0^{L/4} x \sin\frac{n \pi x}{L} dx = - \frac{L}{n \pi} \cdot \frac{L}{4} \cos\frac{n \pi}{4} + \frac{L^2}{(n\pi)^2} \sin\frac{n \pi}{4} - 0