The Lumenary · Early undergraduate

Ordinary differential equations

Name: ______________________

Date: ______________________

You will learn to

Solve separable equations, first-order linear equations (by an integrating factor) and linear equations with constant coefficients
Solve linear systems $x' = Ax$ with the matrix exponential and eigenvalues, and classify planar phase portraits by trace and determinant
State and prove the Picard–Lindelöf theorem, and give examples of non-uniqueness and of finite-time blow-up when its conclusions or hypotheses fail
Analyse equilibria and their stability qualitatively, without solving the equation
Compare exact, qualitative and numerical solutions, and say what kind of evidence each provides

1.
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 $n$ .
2.
Solve $y' + 2y = 4$ with $y(0) = 0$ using an integrating factor. What is $\lim_{t \to \infty} y(t)$ ?
3.
Which is the general real solution of $y'' + 2y' + 5y = 0$ ?
- A. $e^{-t}\big(A\cos 2t + B\sin 2t\big)$
- B. $Ae^{-t} + Be^{2t}$
- C. $e^{t}\big(A\cos 2t + B\sin 2t\big)$
- D. $(A + Bt)e^{-t}$
4.
Classify the equilibrium $0$ of $x' = Ax$ for $A = \begin{pmatrix} 1 & 2 \\ 3 & 2 \end{pmatrix}$ .
- A.A saddle
- B.An unstable node
- C.An unstable spiral
- D.A centre
5.
For which initial value problems does the Picard–Lindelöf theorem guarantee a unique solution near $t = 0$ ? (Select all that apply.)
- A. $x' = \sin x + t^2$ , $x(0) = 0$
- B. $x' = |x|$ , $x(0) = 0$
- C. $x' = \sqrt{|x|}$ , $x(0) = 0$
- D. $x' = x^{1/3}$ , $x(0) = 0$
- E. $x' = x^2$ , $x(0) = 1$
6.
The solution of $x' = x^2$ with $x(0) = \tfrac13$ blows up at a finite time $T > 0$ . Find $T$ .
7.
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 $u \equiv 0$ ?
- A. $(e^{-K(t - t_0)}U)' \le 0$ and $U(t_0) = 0$ , so $U \le 0$ ; with $U \ge 0$ this gives $U \equiv 0$ , hence $u \le KU = 0$ .
- B.Since $u \le KU$ and $U \le KU$ , dividing by $U$ gives $K \ge 1$ , a contradiction.
- C.Differentiate $u \le KU$ to get $u' \le Ku$ , so $u$ is a decaying exponential.
- D.Use the Picard iterates: they converge, so the solution is unique.
8.
Which Lean statement says that $x : \R \to \R$ solves the differential equation $x' = a\,x$ at every time?
- A.∀ t, HasDerivAt x (a * x t) t
- B.HasDerivAt x (a * x 0) 0
- C.∀ t, x t = a * x t
- D.∀ t, HasDerivAt (fun s => a * x s) (x t) t
9.
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.
Computational — Practice

Answer key — Ordinary differential equations

1.
$x(t) = x(0)e^{-kt}$ with $e^{-5730k} = \tfrac12$ . Since $17\,190 = 3 \cdot 5730$ , $x(17\,190) = x(0)\,(e^{-5730k})^3 = x(0)/8$ . So $n = 8$ .
2.
With $P(t) = 2t$ : $(ye^{2t})' = 4e^{2t}$ , so $y e^{2t} = 2e^{2t} - 2$ (using $y(0) = 0$ ), and $y(t) = 2(1 - e^{-2t}) \to 2$ . The limit is the equilibrium of $y' = 4 - 2y$ .
3.
The characteristic equation $r^2 + 2r + 5 = 0$ has roots $-1 \pm 2i$ , so the solutions are $e^{-t}(A\cos 2t + B\sin 2t)$ : an underdamped oscillator.
4.
$\tau = 3$ , $\Delta = 1 \cdot 2 - 2 \cdot 3 = -4 < 0$ : a saddle. Indeed $\lambda^2 - 3\lambda - 4 = (\lambda - 4)(\lambda + 1)$ , with eigenvectors $(2, 3)$ for $\lambda = 4$ and $(1, -1)$ for $\lambda = -1$ .
5.
(a), (b) and (e) have right-hand sides that are Lipschitz in $x$ near the initial point (for (a) and (e) because they are $C^1$ ; for (b) directly). (c) and (d) fail the Lipschitz condition at $x = 0$ , and both really have several solutions.
6.
Separating, $-\frac1x = t + c$ ; $x(0) = \frac13$ gives $c = -3$ , so $x(t) = \frac{1}{3 - t}$ , which tends to $+\infty$ as $t \to 3^-$ . In general $x(0) = x_0 > 0$ blows up at $T = 1/x_0$ : the larger the start, the sooner the explosion.
7.
Option (a). From $U' - KU \le 0$ we get $(e^{-K(t-t_0)}U)' \le 0$ , and $U(t_0) = 0$ , so $e^{-K(t-t_0)}U(t) \le 0$ . But $U \ge 0$ , so $U \equiv 0$ and $u \le KU = 0$ . This is Grönwall's inequality in its simplest form.
8.
∀ t, HasDerivAt x (a * x t) t — exactly the hypothesis of exp_unique in the lesson, which then proves x t = x 0 * exp (a * t).
9.
```
def euler(f, t0, x0, t1, n):
    h = (t1 - t0) / n
    t, x = t0, x0
    for _ in range(n):
        x = x + h * f(t, x)
        t = t + h
    return x
```
For $x' = x$ each step multiplies by $1 + h$ , giving $(1 + 1/n)^n$ — compound interest — whose error is about $e/(2n)$ , so doubling $n$ halves it: Euler's method is first order. For $x' = t$ , two steps give $0 + \tfrac12 \cdot 0 + \tfrac12 \cdot \tfrac12 = \tfrac14$ , an underestimate of the true $\tfrac12$ .