Solving coupled linear systems exactly Problem set A

Write each system in matrix form.

1. (a) \(\dfrac{dx}{dt}=4x+3y\) and \(\dfrac{dy}{dt}=2x-y\) are written as \(\begin{pmatrix}x'\\y'\end{pmatrix}=\mathbf M\begin{pmatrix}x\\y\end{pmatrix}\). Write down \(\mathbf M\).
2. (b) \(\dfrac{dx}{dt}=2y-3x\) and \(\dfrac{dy}{dt}=x\) are written in the same way. Write down \(\mathbf M\).

Use the characteristic equation, then solve for each eigenvector. Any non-zero multiple of an eigenvector is correct.

1. (a) Find the eigenvalues and eigenvectors of \(\mathbf M=\begin{pmatrix}3&1\\2&2\end{pmatrix}\).
2. (b) Find the eigenvalues and eigenvectors of \(\mathbf M=\begin{pmatrix}0&1\\6&1\end{pmatrix}\).

Consider the system \(\dfrac{dx}{dt}=2x+2y\) and \(\dfrac{dy}{dt}=x+3y\). Find the eigenvalues and eigenvectors of the matrix of the system, and write down the general solution.

Consider the system \(\dfrac{dx}{dt}=y\) and \(\dfrac{dy}{dt}=2x+y\). The eigenvalues of the matrix are \(-1\) and \(2\). Given that \(x(0)=5\) and \(y(0)=4\), find \(x(t)\) and \(y(t)\).

Consider the system \(\dfrac{dx}{dt}=x+y\) and \(\dfrac{dy}{dt}=2x\). The eigenvalues of the matrix are \(-1\) and \(2\). Given that \(x(0)=4\) and \(x'(0)=5\), find \(x(t)\).

Two tanks are connected by pipes. After \(t\) hours, tank 1 holds \(x\) kg of salt and tank 2 holds \(y\) kg. The amounts are modelled by \(\dfrac{dx}{dt}=-3x+y\) and \(\dfrac{dy}{dt}=2x-2y\). At the start tank 1 holds 9 kg and tank 2 holds none.

1. (a) Write the system in the form \(\begin{pmatrix}x'\\y'\end{pmatrix}=\mathbf M\begin{pmatrix}x\\y\end{pmatrix}\).
2. (b) Find the eigenvalues and eigenvectors of \(\mathbf M\).
3. (c) Find \(x(t)\) and \(y(t)\).
4. (d) Find the amount of salt in tank 1 after 1 hour, and say what happens to both amounts as \(t\to\infty\).

Consider the system \(\dfrac{dx}{dt}=5x-y\) and \(\dfrac{dy}{dt}=3x+y\) with \(x(0)=6\) and \(y(0)=10\).

1. (a) Find the eigenvalues and eigenvectors of the matrix of the system.
2. (b) Find \(x(t)\) and \(y(t)\).
3. (c) Find \(x\) and \(y\) when \(t=0.5\), to 3 significant figures.