OSA: Ordinary Differential Equations

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

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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.

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

1. Every solution of the differential equation \[ y'(t) = 1 + t^2 + \cos(y(t)) \] is monotonically increasing.
2. 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 \(y'(t) = Ay(t)\).
3. 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.