Partial differential equations and Fourier series

Which function solves the heat equation $u_t = u_{xx}$?

Let $f(x) = 3\sin x - 5\sin 4x$ on $[0, \pi]$. What is its Fourier sine coefficient $b_4 = \frac{2}{\pi}\int_0^\pi f(x)\sin 4x\,dx$?

Solve $u_{tt} = 4u_{xx}$ on $\R$ with $u(0, x) = x^2$ and $u_t(0, x) = 0$. What is $u(1, 0)$?

Let $S_N$ be the partial sums of the Fourier series of the square wave $f = \operatorname{sign}(\sin x)$. Which statements are true? Choose all that apply.

In the proof of the weak maximum principle (Theorem 6), why do we work with $v = u + \varepsilon x^2$ rather than with $u$ itself?

Let u : ℝ → ℝ → ℝ take time first, u t x. Which Lean statement says that u satisfies the heat equation u_t = k u_xx at the point (t, x)?

Apply Parseval's identity to the square wave $f = \operatorname{sign}(\sin x)$, whose sine coefficients are $b_n = \frac{4}{\pi n}$ for odd $n$ and $0$ for even $n$. In the real form, Parseval reads $\frac1\pi\int_{-\pi}^{\pi} f^2\,dx = \frac{a_0^2}{2} + \sum_{n\ge1}(a_n^2 + b_n^2)$. You get $\displaystyle\sum_{n \text{ odd}}\frac{1}{n^2} = \frac{\pi^2}{k}$. What is $k$?

Write sine_coefficients(f, N, M) returning the list [b_1, ..., b_N] of Fourier sine coefficients $b_n = \frac{2}{\pi}\int_0^\pi f(x)\sin(nx)\,dx$ of f on $[0, \pi]$, each computed by the midpoint rule with M subintervals.

On $(-\pi, \pi)$, $x = 2\sum_{n\ge1}(-1)^{n+1}\frac{\sin nx}{n}$ and $x^2 = \frac{\pi^2}{3} + 4\sum_{n\ge1}(-1)^n\frac{\cos nx}{n^2}$. Which statements are true? Choose all that apply.

For $n \ge 1$, $u_n(x, y) = \dfrac{\sin(nx)\sinh(ny)}{n^2}$ solves Laplace's equation with $u_n(x, 0) = 0$ and $\partial_y u_n(x, 0) = \dfrac{\sin nx}{n}$. Which conclusion is correct?