Homework for Part 3
Week 5 : Solve exercises 1 to 3.
Week 6 : Solve exercises 4, 5, 7, 8.
Exercise 1
Consider the equation \(\displaystyle\begin{cases} \dfrac{dN}{dt} = N^2 (1-N^2) \\ N(0) = N_0\in \mathbb{R}\end{cases} .\)
Find the stationary points of this equation.
Study their stability.
Exercise 2
Let \(A = A(t)\) be a \(2\times 2\) matrix. One consider the linear differential system \[\label{eq:1} Y'(t) = A(t) Y(t).\]
If \(Y = \begin{pmatrix} y_1 \\ y_2 \end{pmatrix}\) and \(Z=\begin{pmatrix} z_1 \\ z_2 \end{pmatrix}\) are two solutions of [eq:1], one denotes by \(W(t)\) (so called "Wronskian") the determinant \[W(t) = \begin{vmatrix} y_1(t) & z_1(t) \\ y_2(t) & z_2(t) \end{vmatrix}.\]
Suppose that for some \(t_0\) one has \(W(t_0) = 0.\) Show that \(Y\) and \(Z\) are linearly dependent.
Hint: Use the uniqueness result for the solution of such a system together with an initial value.
Deduce that \(Y,Z\) linearly independent \(\iff \; W(t) \neq 0 \; \forall t.\)
Suppose that \(\displaystyle A = \begin{pmatrix} a_{11}(t) & a_{12} (t) \\ a_{21}(t) & a_{22}(t) \end{pmatrix}.\) Show that \[\label{eq:2} W'(t) = (a_{11}(t) + a_{22} (t)) W(t).\]
Hint: Compute \(W' = (y_1 z_2 - y_2 z_1)'\) and use [eq:1].
Solve the equation [eq:2].
Deduce again that \(W(t_0) = 0 \iff W(t) = 0 \; \forall t.\)
Sketch the generalisation of the results above in dimension \(n.\)
Exercise 3
Consider the differential equation \[y'' +p(t) y' + q(t) y =0.\] Show that this equation is equivalent to a first order system.
Solve it when \(q = 0.\)
Exercise 4
Solve the systems \[y' = \begin{pmatrix} -3 & \sqrt{2} \\ \sqrt{2} & -2 \end{pmatrix} y \qquad , \qquad y' = \begin{pmatrix} 0 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 0 \end{pmatrix} y\]
Exercise 5
Describe the behavior of the solution to \[y' = \begin{pmatrix} 1 & 1 & 2 \\ 0 & 2 & 2 \\ -1 & 1 & 3 \end{pmatrix} y \quad, \quad y(0) = \begin{pmatrix} 2 \\0 \\ 1\end{pmatrix}\] when \(t \to + \infty.\)
Exercise 6
Solve the systems \[y' = \begin{pmatrix} -1 & -1 \\ 2 & -1 \end{pmatrix} y \qquad , \qquad y' = \begin{pmatrix} -3 & 0 & 2 \\ 1 & -1 & 0 \\ -2 & -1 & 0 \end{pmatrix} y.\] Sketch the trajectories when \(t \to + \infty.\)
Exercise 7
Consider a \(2 \times 2\) system with constant coefficients in \(\mathbb{R}\) \[y' = A y.\] Let \(\lambda = \nu + i \mu, \bar{\lambda}\) the eigenvalues of \(A\) and \(\xi, \bar{\xi}\) the corresponding eigenvectors.
Show that if \(\lambda \neq \bar{\lambda}\) then the corresponding eigenvectors are linearly independent.
Show that if \(\xi = a+ib,\) \(a,b\) are linearly independent.
Show that the functions \[a \cos \mu t + b \sin \mu t, b \cos \mu t - a \sin \mu t\] are then linearly independent.
Exercise 8
Show that \(\lambda = 2\) is a triple eigenvalue for the system \[y' = \begin{pmatrix}1 & 1 & 1 \\ 2 & 1 & -1 \\ -3 & 2 & 4 \end{pmatrix}y\] and find three independent solutions to this system.