From 9194e206152ff13b88cd71fd95741b8fa6e901b6 Mon Sep 17 00:00:00 2001
From: Scarlett Gauthier <s.s.gauthier@student.tudelft.nl>
Date: Fri, 28 Aug 2020 09:27:36 +0000
Subject: [PATCH] Finish adding questions for DE1
---
src/differential_equations_1.md | 79 +++++++++++++++++++++++++++++++++
1 file changed, 79 insertions(+)
diff --git a/src/differential_equations_1.md b/src/differential_equations_1.md
index f563d07..f362048 100644
--- a/src/differential_equations_1.md
+++ b/src/differential_equations_1.md
@@ -806,6 +806,85 @@ since we have demonstrated that $**\phi**_{k}(t)$ is a solution of the DE.
(c) $$x^{(2)}_1 x_1 + \dot{x}_1 = 8 x_2$$
$$\dot{x}_2=5tx_2 + x_1$$
+5. [:grinning:] Take the system of equations
+
+ $$\dot{x}_1 = \frac{1}{2} (t-2)x_1 + \frac{1}{2} (t+1)x_2$$
+
+ $$\dot{x}_2 = \frac{1}{2}(t+1)x_1 + \frac{1}{2}(t-2)x_2.$$
+
+ Show that
+
+ $**\Phi**_1(t) = \begin{bmatrix}
+ e^{- \frac{3}{2} t} \\
+ -e^{- \frac{3}{2} t} \\
+ \end{bmatrix}$ and $**\Phi**_2(t)=\begin{bmatrix}
+ e^{\frac{1}{2}(t^2-t)} \\
+ e^{\frac{1}{2}(t^2-t)} \\
+ \end{bmatrix}$
+
+ constitute a basis for the solution space of this system of equations.
+ To this end, first verify that they are indeed solutions and then that
+ they form a basis.
+
+6. [:grinning:] Take the system of equations
+
+ $$\dot{x}_1=x_2$$
+
+ $$\dot{x}_2=x_2.$$
+
+ Re-write this system of equations into the general form
+
+ $$\dot{**x**} = **A** **x**$$
+
+ and then find the general solution. Specify the general solution for the
+ following initial conditions
+
+ (a) $**x**(0) = \begin{bmatrix}
+ 1 \\
+ 0 \\
+ \end{bmatrix}$
+
+ (b) $**x**(0) = \begin{bmatrix}
+ 0 \\
+ 1 \\
+ \end{bmatrix}$
+
+7. [:smirk:] Find the general solution of
+
+ $$\begin{bmatrix}
+ \dot{x}_1 \\
+ \dot{x}_2 \\
+ \dot{x}_3 \\
+ \end{bmatrix} = \begin{bmatrix}
+ 2 & 0 & 2 \\
+ 0 & 2 & 0 \\
+ 1 & 0 & 3 \\
+ \end{bmatrix} \begin{bmatrix}
+ x_1 \\
+ x_2 \\
+ x_3 \\
+ \end{bmatrix}.$$
+
+ Then, specify the solution for the initial conditions
+
+ (a) $\begin{bmatrix}
+ 0 \\
+ 0 \\
+ 1 \\
+ \end{bmatrix}$
+
+ (b) $\begin{bmatrix}
+ 1 \\
+ 0 \\
+ 0 \\
+ \end{bmatrix}$
+
+8. [:sweat:] *Bonus question - A question of this kind will not be on the exam*
+
+ Show that $e^{**A** t} e^{-**A** t} = \mathbbm{1}$, where $**A**$ is an
+ $n \times n$ matrix and $\mathbbm{1}$ is the $n \times n$ identity matrix,
+ by using the definition of the matrix exponential in terms of the Taylor series!
+
--
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