1. **Problem statement:** A rod of length $l$ with insulated sides is initially at a uniform temperature $u_0$. Its ends are suddenly cooled to 0 degrees Celsius and kept at that temperature. We need to find the temperature function $u(x,t)$ describing the temperature at position $x$ and time $t$. 2. **Set up the heat equation:** The temperature distribution in the rod satisfies the one-dimensional heat equation: $$\frac{\partial u}{\partial t} = \alpha^2 \frac{\partial^2 u}{\partial x^2}$$ where $\alpha^2$ is the thermal diffusivity constant. 3. **Boundary conditions:** Since the ends are kept at 0 degrees, $$u(0,t) = 0, \quad u(l,t) = 0 \quad \text{for all } t > 0$$ 4. **Initial condition:** The initial temperature is uniform, $$u(x,0) = u_0 \quad \text{for } 0 < x < l$$ 5. **Method of solution:** Use separation of variables by assuming $$u(x,t) = X(x)T(t)$$ Substitute into the heat equation and separate variables to get eigenvalue problems. 6. **Spatial part:** Solve $$X'' + \lambda X = 0$$ with boundary conditions $X(0)=0$ and $X(l)=0$. The eigenvalues and eigenfunctions are $$\lambda_n = \left(\frac{n\pi}{l}\right)^2, \quad X_n(x) = \sin\left(\frac{n\pi x}{l}\right), \quad n=1,2,3,...$$ 7. **Temporal part:** For each $n$, $$T_n(t) = e^{-\alpha^2 \lambda_n t} = e^{-\alpha^2 \left(\frac{n\pi}{l}\right)^2 t}$$ 8. **General solution:** The temperature is a sum over all modes, $$u(x,t) = \sum_{n=1}^\infty b_n \sin\left(\frac{n\pi x}{l}\right) e^{-\alpha^2 \left(\frac{n\pi}{l}\right)^2 t}$$ 9. **Find coefficients $b_n$ using initial condition:** $$u(x,0) = u_0 = \sum_{n=1}^\infty b_n \sin\left(\frac{n\pi x}{l}\right)$$ Multiply both sides by $\sin\left(\frac{m\pi x}{l}\right)$ and integrate from 0 to $l$: $$\int_0^l u_0 \sin\left(\frac{m\pi x}{l}\right) dx = b_m \frac{l}{2}$$ 10. **Calculate $b_n$:** $$b_n = \frac{2}{l} \int_0^l u_0 \sin\left(\frac{n\pi x}{l}\right) dx = \frac{2 u_0}{l} \left[-\frac{l}{n\pi} \cos\left(\frac{n\pi x}{l}\right)\right]_0^l = \frac{2 u_0}{n\pi} (1 - (-1)^n)$$ 11. **Simplify $b_n$:** For even $n$, $b_n=0$; for odd $n$, $$b_n = \frac{4 u_0}{n\pi}$$ 12. **Final temperature function:** $$u(x,t) = \sum_{n=1,3,5,...}^\infty \frac{4 u_0}{n\pi} \sin\left(\frac{n\pi x}{l}\right) e^{-\alpha^2 \left(\frac{n\pi}{l}\right)^2 t}$$ This series describes the temperature distribution in the rod over time.