DP Mathematics: Applications and Interpretation Questionbank

Determine, with justification, whether the equilibrium point $(0, 0)$ is stable or unstable.

Hence, write down the general solution of the system.

Find the eigenvalues and corresponding eigenvectors of the matrix $\left(\begin{array}{cc}-4& 0\\ 3& -2\end{array}\right)$.

(i)   at $(4, 0)$.

(ii)  at $(-4, 0)$.

Sketch a phase portrait for the general solution to the system of coupled differential equations for $-6\le x\le 6$, $-6\le y\le 6$.

Use Euler’s method to sketch the trajectory of the phase portrait, for $4\le M\le 7$ and $1.5\le S\le 3$, over the first nine years.

$\gamma MS$

Toxic sewage is added to the Mediterranean Sea. Alessia claims this reduces the shark population growth rate and hence the value of $\gamma$ is halved. No other parameter changes.

Using your phase portrait, or otherwise, determine whether the mackerel population density would be sufficient to support sustainable fishing during the first nine years.

$\delta S$

show that, at the equilibrium point, the value of the mackerel population density is $\frac{\delta }{\gamma }$;

State two reasons why Alessia’s conclusion, found in part (f)(ii), might not be valid.

find the value of the shark population density at the equilibrium point.

Global warming increases the temperature of the Mediterranean Sea. Alessia claims that this promotes the mackerel population growth rate and hence the value of $\alpha$ is doubled. No other parameter changes.

the eigenvectors.

Find the population of $\text{Y}$ at this value of $t$. Give your answer to the nearest $10$ herbivores.

Sketch the phase portrait for this system, for $x, y>0$.

On your sketch show

• the equation of the line defined by the eigenvector in the first quadrant
• at least two trajectories either side of this line using arrows on those trajectories to represent the change in populations as t increases

Find the value of $t$ at which $x=0$.

the eigenvalues.

Write down a condition on the size of the initial population of $\text{Y}$ if it is to avoid its population reducing to zero.

Hence write down the general solution of the system of equations.

Find $A$ in terms of $p$ and $q$.

Show that $A=\frac{192}{q}+3q+48$.

Find $\frac{dA}{dq}$.

Hence, or otherwise, find this value of $q$.

Write down an equation Ellis could solve to find this value of $q$.

Given that $y=\dot{x}$, show that $\dot{y}=-1.25x-3y$.

Find the eigenvalues of matrix $\mathit{A}$.

Write down the matrix $\mathit{A}$.

Find the eigenvectors of matrix $\mathit{A}$.

Given that when $t=0$ the shock absorber is displaced $8 \text{cm}$ and its velocity is zero, find an expression for $x$ in terms of $t$.

In the case when there is a pair of purely imaginary eigenvalues you are told that the solution will form an ellipse.  You are also told that the initial conditions are such that the ellipse is large enough that it will cross both the positive and negative $x$ axes and the positive and negative $y$ axes.

By considering the differential equations at these four crossing point investigate if the trajectory is in a clockwise or anticlockwise direction round the ellipse. Give your decision in terms of $b$ and $c$. Using part (b) (ii) show that your conclusions are consistent.

Show that if the system has two distinct real eigenvalues then ${\left(a-d\right)}^2+4bc>0$.

Explain why $b$ and $c$ must have opposite signs.

Use the eigenvalue method to find the general solution to this system of equations.

Find two conditions that must be satisfied by $a$, $b$, $c$, $d$.

Find the eigenvalues for the matrix $\left(\begin{array}{cc}0 & 1\\ -6 & -5\end{array}\right)$.

Hence state the long-term velocity of the particle.

Use the substitution $y=\frac{dx}{dt}$ to show that this equation can be written as

$\left(\begin{array}{c}\frac{dx}{dt}\\ \frac{dy}{dt}\end{array}\right)=\left(\begin{array}{cc}0 & 1\\ -6 & -5\end{array}\right)\left(\begin{array}{c}x\\ y\end{array}\right)$.

As $t\to \mathrm{\infty }$, find an asymptote to the trajectory of the particular solution found in (b)(i) and state if this trajectory will be moving towards or away from the origin.

Hence find the solution when $t=1$.

Given the initial conditions that when $t=0$, $x=150$, $y=50$, find the particular solution.

Find the value of $\frac{\text{d}y}{\text{d}x}$ when $t=0$.

On the following axes, sketch a possible trajectory for the growth of the two fungi, making clear any asymptotic behaviour.

Hence, verify that $\left(\begin{array}{c}x\\ y\end{array}\right)=\left(\begin{array}{c}1\\ -1\end{array}\right)e^{2t}$ is a solution to the above system.

 Find the equation of this asymptote.

Show how using the substitution $X=x-10\text{,}Y=y-20$ transforms the system of differential equations into $\begin{array}{c}\dot{X}=X+2Y\\ \dot{Y}=2X+Y\end{array}$.

With the initial conditions as given in part (b) state if the equilibrium point is stable or unstable.

Use part (f) to find limit of the ratio $\frac{y}{x}$ as $t\to \mathrm{\infty }$.

State the direction of the trajectory, including the quadrant it is in as it approaches this asymptote.

Find the equilibrium point for this system.

If initially $x=100$ and $y=50$ use Euler’s method with an time increment of 0.1 to find an approximation for the values of $x$ and $y$ when $t=1$.

If initially at $t=0\text{,}x=20\text{,}y=10$ find the particular solution.

Find the values of $x$ and $y$ when $t=2$.

If instead the initial conditions were given as $x=20$ and $y=10$, find the particular solution for $\left(\begin{array}{c}x\\ y\end{array}\right)$ of the original system, in this case.

Extend this method to conjecture the limit of the ratio $\frac{y}{x}$ as $t\to \mathrm{\infty }$.

Show that the matrix  has (sadly) only one eigenvalue.  Find this eigenvalue and an associated eigenvector.

Find the particular solution to the original system, given the initial conditions of part (b).

Solve this system of equations by the eigenvalue method and hence find the general solution for $\left(\begin{array}{c}x\\ y\end{array}\right)$ of the original system.

Hence find the exact values of $x$ and $y$ when $t=1$, giving the answers to 4 significant figures.

With the initial conditions as given in part (j), determine if the equilibrium point is stable or unstable.

Verify that $\left(\begin{array}{c}x\\ y\end{array}\right)=\left(\begin{array}{c}t\\ -t+1\end{array}\right)e^{2t}$ is also a solution.

Hence find the exact solution of the second order differential equation.

Show that the system of coupled first order equations:

$\frac{dx}{dt}=y$

$\frac{dy}{dt}=-2x-3y$

can be written as the given second order differential equation.