24. The solution $u(x, t)$ of the heat transfer problem \begin{equation*} \begin{cases} 2 \frac{\partial^2 u}{\partial x^2} = \frac{\partial u}{\partial t} & 0 < x < \pi, t > 0 \\ u(0, t) = 0 & u(\pi, t) = 0 \\ u(x, 0) = \sin(x) + \frac{1}{3} \sin(3x) & 0 < x < \pi \end{cases} \end{equation*} is \begin{align*} \text{A. } & u(x, t) = \sin(x) e^{-2t} + \frac{1}{3} \sin(3x) e^{-18t} \\ \text{B. } & u(x, t) = \sin(x) e^{-2t} + \frac{1}{3} \sin(2x) e^{-8t} \\ \text{C. } & u(x, t) = \sin(x) e^{-t} + \frac{1}{3} \sin(3x) e^{-9t} \\ \text{D. } & u(x, t) = \sin(x) e^{-2t} + \sin(3x) e^{-18t} \\ \text{E. } & u(x, t) = \sin(x) e^{-2t} + \sin(2x) e^{-8t} + \sin(3x) e^{-18t} \end{align*}
