5.16 Euler’s method questions for IB Maths AI

Find the equation of the tangent to the solution curve at $P$.

Use your tangent to estimate the value of $y$ on this solution curve when $x=1.2$.

By considering ${\displaystyle \frac{{\mathrm{d}}^{2}y}{\mathrm{d}{x}^2}}$ at $P$, state whether your estimate in part (b) is an underestimate or an overestimate. Justify your answer.

On the diagram, sketch the solution curve through $P$.

Find the values of $P$ for which ${\displaystyle \frac{\mathrm{d}P}{\mathrm{d}t}}=0$.

Interpret, in context, the non-zero value of $P$ found in part (a).

Find the number of rabbits when the population is increasing most rapidly, and the rate of increase at that moment.

Use Euler's method with a step length of 0.5 to estimate the number of rabbits after 3 years.

The solution of the differential equation is $P={\displaystyle \frac{1200}{1+11{\mathrm{e}}^{-0.4t}}}$. Find the percentage error in the estimate in part (d).

Explain why Euler's method underestimates the number of rabbits for $0<t\le 3$.

Find the time at which the population is increasing most rapidly.

At one moment there are 6000 aphids and 100 ladybirds in the greenhouse.

Find the value of $\frac{\mathrm{d}x}{\mathrm{d}t}$ and the value of $\frac{\mathrm{d}y}{\mathrm{d}t}$ at this moment, and interpret your answers in context.

Consider the case in which there are no ladybirds in the greenhouse.

Write down the differential equation satisfied by $x$.

Hence find the long-term number of aphids when there are no ladybirds, justifying your answer.

Show that $(4,2)$ is an equilibrium point of the system.

Write down the coordinates of the other two equilibrium points.

To investigate the behaviour close to $(4,2)$, let $x=4+p$ and $y=2+q$, where $p$ and $q$ are small. You may assume that terms in $p^2$, $q^2$ and $pq$ can be ignored.

Show that $\frac{\mathrm{d}p}{\mathrm{d}t}\approx -0.4p-1.6q$.

Find a similar approximation for $\frac{\mathrm{d}q}{\mathrm{d}t}$.

The approximations can be written as $\left(\begin{array}{c}\dot{p}\\ \dot{q}\end{array}\right)=\mathit{M}\left(\begin{array}{c}p\\ q\end{array}\right)$.

Find the eigenvalues of $\mathit{M}$.

Hence describe the behaviour of the aphid and ladybird populations when they start close to $(4,2)$.

At the start of one season there are 2000 aphids and 100 ladybirds in the greenhouse.

Use Euler's method with a step length of 0.25 weeks to estimate the number of aphids and the number of ladybirds after 1 week.

Sketch, on a set of axes with $x$ on the horizontal axis and $y$ on the vertical axis, the trajectory that starts at $(2,1)$. Show the direction of motion along the trajectory.

Selin considers spraying a pesticide that kills aphids and ladybirds at the same rate. She modifies her model to

Find the equilibrium point of the modified model at which both species are present.

Comment on whether spraying the pesticide is a sensible way to reduce the aphid population in the long term.

Use Euler's method with a step length of 0.1 to find an approximate value of $y$ when $x=0.2$. Show all your working.

By considering ${\displaystyle \frac{{\mathrm{d}}^{2}y}{\mathrm{d}{x}^2}}$, explain why your answer to part (a) is likely to be an underestimate.

Explain why Euler's method cannot be used to approximate the solution that passes through the point $(1,0)$.

Use Euler's method with a step length of 0.25 to find an approximate value of $y$ when $x=1$, given that $y=2$ when $x=0$.

Solve the differential equation to find the exact solution through $(0,2)$, giving $y$ in terms of $x$.

Find the percentage error in the approximation found in part (b).

State how the approximation in part (b) could be improved.

Show that the solution curve through $(0,2)$ never meets the $x$-axis.

By letting $\omega =\frac{\mathrm{d}\theta }{\mathrm{d}t}$, write the differential equation as a system of first-order equations in the form $\left(\begin{array}{c}\dot{\theta }\\ \dot{\omega }\end{array}\right)=\mathit{M}\left(\begin{array}{c}\theta \\ \omega \end{array}\right)$, where $\mathit{M}$ is a $2\times 2$ matrix in terms of $c$.

The closer is first set with $c=5$.

Show that the eigenvalues of $\mathit{M}$ are $-1$ and $-4$.

Find an eigenvector for each eigenvalue.

Hence show that $\theta =1.6{\mathrm{e}}^{-t}-0.4{\mathrm{e}}^{-4t}$.

The door is considered shut once $\theta <0.05$.

Find the time taken for the door to be shut.

Show that, with this setting, the door never swings past its closed position.

Sketch the phase portrait for this system, with $\theta$ on the horizontal axis and $\omega$ on the vertical axis. Your sketch should show the eigenvector lines and the trajectory that starts at $(1.2,0)$.

After some years the closer wears and the damping setting falls to $c=2$.

Find the eigenvalues of $\mathit{M}$ for this setting.

Hence describe the motion of the door with this setting.

Use Euler's method with a step length of 0.1 seconds to estimate the value of $\theta$ when $t=0.5$ for the worn closer.

A computer solution of the differential equation gives $\theta =0.7916$ when $t=0.5$.

Calculate the percentage error in your answer to part (h).

Find the set of values of $c$ for which $\mathit{M}$ has two distinct real eigenvalues.

Explain why the manufacturer would not choose a very large value of $c$.