Ordinary differential equations

Carbon-14 decays according to $x' = -kx$, with a half-life of about $5730$ years. A piece of charcoal from an ancient hearth is $17\,190$ years old. Its carbon-14 content is $1/n$ of the original amount. Find $n$.

Solve $y' + 2y = 4$ with $y(0) = 0$ using an integrating factor. What is $\lim_{t \to \infty} y(t)$?

Which is the general real solution of $y'' + 2y' + 5y = 0$?

Classify the equilibrium $0$ of $x' = Ax$ for $A = \begin{pmatrix} 1 & 2 \\ 3 & 2 \end{pmatrix}$.

For which initial value problems does the Picard–Lindelöf theorem guarantee a unique solution near $t = 0$? (Select all that apply.)

The solution of $x' = x^2$ with $x(0) = \tfrac13$ blows up at a finite time $T > 0$. Find $T$.

In the uniqueness proof, $u(t) = |x(t) - y(t)|$ satisfies $u(t) \le K\int_{t_0}^t u(s)\,ds$ for $t \ge t_0$, and $U(t) = \int_{t_0}^t u$. Which step finishes the proof that $u \equiv 0$?

Which Lean statement says that $x : \R \to \R$ solves the differential equation $x' = a\,x$ at every time?

Write euler(f, t0, x0, t1, n) implementing Euler's method $x_{k+1} = x_k + h f(t_k, x_k)$ with $n$ equal steps of size $h = (t_1 - t_0)/n$, returning the approximation of $x(t_1)$. Check that for $x' = x$, $x(0) = 1$ it returns $(1 + 1/n)^n$, and that doubling $n$ roughly halves the error.