Coupled & Second Order Differential Equations (DP IB Applications & Interpretation (AI): HL): Exam Questions

Consider the following system of coupled differential equations

 $x=-3x+e^{-4t}y$

$y=6e^{-2t}x+y$

with the initial condition $x=1,y=2$ when $t=0$.

Use the Euler method with a step size of 0.1 to find approximations for the values of $x$  and $y$ when $t=0.5$.        

Show that the system has no equilibrium points other than the origin, for any value of $t$.

Consider the following system of differential equations:

 $\frac{dx}{dy}=\frac{1}{2}x-2y$

$\frac{dy}{dx}=x-\frac{5}{2}y$ 

By first finding the eigenvalues and corresponding eigenvectors of an appropriate matrix, determine the general solution of the system.

When $t=0,x=-3$ and $y=2$.

Use the given initial condition to determine the exact solution of the system.

Describe the long-term behaviour of the variables $x$ and $y$.

The rates of change of two variables, $x$ and $y$, are described by the following system of differential equations:

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

$\frac{dy}{dt}=3x-4y$ 

The matrix  $\left(\begin{array}{cc}3& -2\\ 3& -4\end{array}\right)$  has eigenvectors $\left(\begin{array}{c}2\\ 1\end{array}\right)$  and $\left(\begin{array}{c}1\\ 3\end{array}\right)$.  Initially  $x=7$ and $y=1$. 

Use the above information to find the exact solution to the system of differential equations.

Use the Euler method with a step size of 0.2 to find approximations for the values of $x$  and $y$ when $t=1$.

(i) Find the percentage error of the approximations from part (b) compared with the exact values of $x$ and $y$ when $t=1$. 

(ii) Explain how the approximations found in part (b) could be improved.

Consider a system of coupled differential equations with a general solution given by

 $x=A{e}^{pt}\left(\begin{array}{c}2\\ 1\end{array}\right)+B{e}^{qt}\left(\begin{array}{c}-2\\ 3\end{array}\right)$ 

where $p$ and $q$ are real constants. 

For each of the relationships between $p$ and $q$ given below,

i) sketch the phase portrait for the system 

ii) state whether the point  is a stable equilibrium point or an unstable equilibrium point.

$p<q<0.$

$p<0<q$

$0<p<q$

The behaviour of two variables, $x$and $y$, is modelled by the following system of differential equations:

$\frac{dx}{dt}=3x-5y$           $\frac{dy}{dt}=x-y$

where $x=1$ and  $y=1$ when $t=0$. 

The matrix $\left(\begin{array}{cc}3& -5\\ 1& -1\end{array}\right)$ has eigenvalues of  $1+\mathrm{i}$ and $1-\mathrm{i}$. 

(i) Find the values of  $\frac{dx}{dt}$ and $\frac{dy}{dt}$ at the point $\left(0,1\right)$. 

(ii) Hence sketch the phase portrait of the system with the given initial condition.

It is suggested that the variables might better be described by the system

 $\frac{dx}{dy}=-3x-5y$       $\frac{dy}{dt}=x+y$ 

with the same initial conditions. 

Calculate the eigenvalues of the matrix  $\left(\begin{array}{cc}-3& -5\\ 1& 1\end{array}\right)$

Hence describe how your phase portrait from part (a)(ii) would change to represent this new system of differential equations.

Scientists have been tracking levels, $x$ and $y$, of two atmospheric pollutants, and recording the levels of each relative to historical baseline figures (so a positive value indicates an amount higher than the baseline and a negative value indicates an amount less than the baseline).  Based on known interactions of the pollutants with each other and with other substances in the atmosphere, the scientists propose modelling the situation with the following system of differential equations:

$\frac{dx}{dt}=x-2y$

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

Find the values of $\frac{dx}{dt}$ and $\frac{dy}{dt}$ at the points  $\left(1,0\right)$ and $\left(0,1\right)$.

Find the eigenvalues of the matrix  $\left(\begin{array}{cc}1& -2\\ 1& -1\end{array}\right)$

At the start of the study both pollutants are above baseline levels, with $x=5$ and $y=3$.

Use the above information to sketch a phase portrait showing the long-term behaviour of $x$ and $y$.

Scientists are studying populations of a prey species and a predator species within a particular region.  They initially model the two species by the system of differential equations  $\frac{dx}{dt}=1.9x-0.2y$ and $\frac{dy}{dt}=0.3x+2.6y$ , where $x$ represents the size of the prey population (in thousands) and $y$ represents the size of the predator population (in hundreds).  Initially there are 2000 animals in the prey population and 450 in the predator population. 

Given that the eigenvalues of the matrix $\left(\begin{array}{cc}1.9& -0.2\\ 0.3& 2.6\end{array}\right)$ are $2.5$ and 2, with corresponding eigenvectors $\left(\begin{array}{c}-1\\ 3\end{array}\right)$ and $\left(\begin{array}{c}-2\\ 1\end{array}\right)$, sketch a possible trajectory for the change in the populations of the two animals over time.

Research suggests that neither species will disappear from the region in the foreseeable future. 

Criticise the model above, particularly in light of this research result.

It is suggested that the system of equations  $\frac{dx}{dt}=\left(10-2y\right)x$ and  $\frac{dy}{dt}=\left(3x-6\right)y$ should be used as a model instead, where $t$ is measured in decades (1 decade= 10 years ).

Determine the equilibrium points for the system under this model.

(i) Use the Euler method with a step size of 0.002 to find approximations for the values of $x$ and $y$ at one-year intervals up to 8 years after the start of the study.

(ii) Use the values from (d)(i) to sketch a possible trajectory for the change in the populations of the two animals over time, and state what this suggests about the long-term behaviour of the two animal populations under the revised model.

A particle moves in a straight line, such that its displacement $x$ metres at time $t$ seconds is described by the differential equation

 $\frac{{\mathrm{d}}^{2}x}{\mathrm{d}{t}^2}+7\frac{dx}{dt}+13x=109$ 

where $\frac{dx}{dt}$ and $\frac{{\mathrm{d}}^{2}x}{\mathrm{d}{t}^2}$ represent the particle’s velocity and acceleration respectively.

By letting  $y=\frac{dx}{dt}$,  show that the differential equation above can be written as a system of first order differential equations.

When $t=0$,  the displacement of the particle is zero and the velocity is $-2{\mathrm{ms}}^{-1}$.

By applying Euler’s method with a step size of 0.1 to the system of equations found in part (a), along with the given initial condition, find approximations for the

i) displacement

ii) velocity

of the particle at time t = 0.5 .

Use the Euler method to determine the long-term stable value of the particle’s displacement.

Use your answer from part (c) to explain why the long-term stable value of the particle’s velocity must be zero.

Consider the following system of coupled differential equations

 $\dot{x}=x^2+2ty$

$\dot{y}=-6x+y+14t$ 

with the initial condition $x=-2,y=-12$ when $t=0.$ 

Use the Euler method with a step size of 0.1 to find approximations for the values of $x$  and $y$ when $t=0.5.$        

(i) Show that the system has a single equilibrium point at time $t=0$ and write down its coordinates.

(ii) Find the coordinates of the equilibrium points in terms of $t$. Hence show that, for times $t>0$, the system has no equilibrium points at which the values of both $x$ and $y$ are non-negative.

Consider the following system of differential equations:

$\frac{dx}{dt}=-\frac{23}{14}x+\frac{5}{7}y$

$\frac{dy}{dt}=\frac{15}{14}x+\frac{1}{7}y$ 

Given that  $x=8$  and $y=3$ when $t=0$,  

use a matrix method to determine the exact solution of the system.

Hence determine the long-term ratio of the value of $x$ to the value of $y$.

The rates of change of two variables, $x$ and $y$, are described by the following system of differential equations:

$\frac{dx}{dt}=-0.3x-2.1y$        $\frac{dy}{dt}=-8.1x+3.3y$ 

The matrix $\left(\begin{array}{cc}-0.3& -2.1\\ -8.1& 3.3\end{array}\right)$ has eigenvalues -3 and 6.  Initially $x=0$  and $y=3$.

Use the above information to find the exact solution to the system of differential equations.

Use the Euler method with a step size of 0.2 to find approximations for the values of $x$ and $y$ when  $t=1$.

Compare the ratio of the approximations from part (b) with the ratio of $x$ to $y$ that you would expect in the long term based on your answer to part (a).

(i) Find the percentage error of the approximations from part (b) compared with the exact values of $x$ and $y$ when $t=1$.

(ii) Explain how the approximations found in part (b) could be improved.

Given that the matrices $\left(\begin{array}{cc}-0.2& 2.4\\ 2.4& 1.2\end{array}\right),\left(\begin{array}{cc}1.36& 0.48\\ 0.48& 1.64\end{array}\right)$ and $\left(\begin{array}{cc}-2.28& 0.96\\ 0.96& -1.72\end{array}\right)$ all have $\left(\begin{array}{c}3\\ 4\end{array}\right)$ and $\left(\begin{array}{c}-4\\ 3\end{array}\right)$ as eigenvectors,

i) sketch the phase portrait, and

ii) state whether the point $\left(0,0\right)$ is a stable equilibrium point or an unstable equilibrium point 

for each of the systems of differential equations given below.

$\frac{dx}{dt}=-0.2x+2.4y\frac{dy}{dt}=2.4x+1.2y$

$\frac{dx}{dt}=1.36x+0.48y\frac{dy}{dt}=0.48x+1.64y$

$\frac{dx}{dt}=-2.28x+0.96y\frac{dy}{dt}=0.96x-1.72y$

The behaviour of two variables, $x$ and $y$, is modelled by the following system of differential equations:

 $\frac{dx}{dt}=-x-10y\frac{dy}{dt}=10.1x-3y$ 

where $x=1$ and $y=1$ when $t=0.$  

Sketch the phase portrait of the system with the given initial condition.

Now consider instead the following system of differential equations

$\frac{dx}{dt}=x-10y\frac{dy}{dt}=10.1x+3y$

with the same initial conditions.

Describe briefly how the phase portrait for this system would differ from the phase portrait drawn in part (a).  Be sure to justify your answer.

The amounts, $x$ and $y$, of two reactive chemicals in a solution are modelled by the following system of differential equations:

 $\frac{dx}{dy}=5x-8.5y+9.5\frac{dy}{dt}=4x-5y-5$ 

Find the equations of the lines on which (i) $\frac{dx}{dt}$ and (ii) $\frac{dy}{dt}$ are equal to zero, and hence determine the equilibrium point of the system.

Use the substitutions $u=x-10$ and $v=y-7$ to rewrite the equations as a system of coupled differential equations in $u$ and $v$.

Determine the nature of the solution trajectories for the system of equations in $u$ and $v$  found in part (b).

Initially $x=1$ and $y=1$.

Use your answers to parts (a) and (c) to sketch a phase portrait showing the long-term behaviour of $x$ and $y$.  You may take as a given that both $x$ and $y$ remain non-negative for all values of $t$.

Scientists are studying populations of a prey species and a predator species within a particular ecosystem.  They model the two populations by the system of equations

 $\frac{dx}{dt}=\left(a-by\right)x\frac{dy}{dt}=\left(cx-d\right)y$ 

where $x$ represents the size of the prey population,   $y$ represents the size of the predator population, and $a,b,c$ and $d$ are all positive real parameters.

Write down (i) the coordinates of the equilibrium points of the system, and (ii) the equations of the lines on which the (local) minimum and maximum values of $x$ and $y$ will be located.  Give your answers in terms of $a,b,c$ and $d$ as appropriate.

Parameter  is sometimes referred to as the ‘prey population growth parameter’, while parameter $d$ is sometimes referred to as the ‘predator population extinction parameter’. 

Using mathematical reasoning, explain briefly (i) why those names are suitable for parameters $a$ and $b$, and (ii) what parameters $b$ and $c$ represent in the model.

Let $a=13,b=5,c=4$ and  $d=9$,  with $x=1$ and $y=1$ when $t=0$.

By first using the Euler method with a step size of 0.002 to find approximations for the values of $x$ and $y$ between $t=0$ and $t=0.64$,  sketch a phase portrait showing an approximate solution trajectory for the system with the given parameter values and initial conditions.

A particle moves in a straight line, such that its displacement $x$ metres at time $t$ seconds is described by the differential equation

 $\stackrel{"}{x}+5\dot{x}+6x=0$ 

Show that the second order differential equation above can be rewritten as a system of coupled first order differential equations.

When $t=0$ ,  the displacement of the particle is $-3$ m and the velocity is  $-10{\mathrm{ms}}^{-1}$.

By applying Euler’s method with a step size of 0.1 to the system of equations found in part (a), find approximations for (i) the displacement and (ii) the velocity of the particle at time $t=0.5$.

By first finding the exact solution to the system of equations found in part (a), determine the percentage error of the values for the displacement and velocity at time $t=0.5$ that were found in part (b).

Sketch the trajectory of your exact solution from part (c) on a phase diagram, showing the relationship between the particle’s displacement and velocity as time t increases.