Solving coupled linear systems exactly Problem set B

Write each system in matrix form.

1. (a) \(\dfrac{dx}{dt}=3x-5y\) and \(\dfrac{dy}{dt}=6x+2y\) are written as \(\begin{pmatrix}x'\\y'\end{pmatrix}=\mathbf M\begin{pmatrix}x\\y\end{pmatrix}\). Write down \(\mathbf M\).
2. (b) \(\dfrac{dx}{dt}=y-4x\) and \(\dfrac{dy}{dt}=2x\) 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 \(\mathbf M=\begin{pmatrix}4&1\\2&3\end{pmatrix}\).
2. (b) Find the eigenvalues and eigenvectors of \(\mathbf M=\begin{pmatrix}1&1\\4&-2\end{pmatrix}\).

Consider the system \(\dfrac{dx}{dt}=-2x+2y\) and \(\dfrac{dy}{dt}=2x+y\). 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 \(-2\) and \(1\). Given that \(x(0)=6\) and \(y(0)=3\), find \(x(t)\) and \(y(t)\).

Consider the system \(\dfrac{dx}{dt}=x+2y\) and \(\dfrac{dy}{dt}=3x\). The eigenvalues of the matrix are \(-2\) and \(3\). Given that \(x(0)=4\) and \(x'(0)=2\), find \(x(t)\).

A drug is injected into the blood. After \(t\) hours, the blood holds \(x\) mg of the drug and body tissue holds \(y\) mg. The amounts are modelled by \(\dfrac{dx}{dt}=-3x+y\) and \(\dfrac{dy}{dt}=4x-3y\). At the start the blood holds 8 mg and the tissue 4 mg.

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 the drug in the blood after 1 hour, and say what happens to both amounts as \(t\to\infty\).

Consider the system \(\dfrac{dx}{dt}=4x+y\) and \(\dfrac{dy}{dt}=3x+2y\) with \(x(0)=6\) and \(y(0)=2\).

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.