# OSA: Ordinary Differential Equations

## Question 1

Solve the following system of linear differential equations: \[ y'(t) = \begin{pmatrix}2&1\\6&1\end{pmatrix}y(t) + \begin{pmatrix}0\\e^{2t}\end{pmatrix} \quad\text{for }t \geq 0\quad\text{with}\quad y(0) = \begin{pmatrix}1\\0\end{pmatrix}. \]

*Note:* Since this question requires you to provide a rigorous justification of your claims (i.e. a proof), your answer can not be checked automatically. We invite you to send your solutions to this email address and will get back to you with feedback.

## Question 2

Show that every solution of the differential equation \[ y''(t) = y'(t) + \sin(y(t)) \] is smooth (i.e. it has derivatives of all orders).

*Note:* Since this question requires you to provide a rigorous justification of your claims (i.e. a proof), your answer can not be checked automatically. We invite you to send your solutions to this email address and will get back to you with feedback.

## Question 3

Decide which of the following statements are correct and briefly justify your answer or give a counterexample.

- Every solution of the differential equation \[ y'(t) = 1 + t^2 + \cos(y(t)) \] is monotonically increasing.
- The function \[ t \mapsto e^{tA}\cdot \mathbf{v} \] with matrix \(A \in \mathbf{R}^{3\times 3}\) and vector \(\mathbf{v} = (1, -1, -1)^\mathsf{T} \in \mathbf{R}^3\) satisfies \(y'(t) = Ay(t)\).
- If \(f : \mathbf{R}^2 \to \mathbf{R}\) is a continuous function and \(u : \mathbf{R} \to \mathbf{R}\) a solution to \(y'(t) = f(t,y(t))\) with \(y(1) = 0\), then \[ u(t) = \int_1^t f(s,u(s))\,\mathrm{d}s \] for all \(t \in \mathbf{R}\).

*Note:* Since this question requires you to provide a rigorous justification of your claims (i.e. a proof), your answer can not be checked automatically. We invite you to send your solutions to this email address and will get back to you with feedback.