5.18 Second-order differential equations questions for IB Maths AI

By letting $y={\displaystyle \frac{\mathrm{d}x}{\mathrm{d}t}}$, write the differential equation as a system of two first-order differential equations.

Use Euler's method with a step length of 0.1 to estimate the displacement of the mass when $t=0.5$.

By letting $y={\displaystyle \frac{\mathrm{d}x}{\mathrm{d}t}}$, show that the equation can be written as

where $\mathit{A}$ is a $2\times 2$ matrix to be written down.

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

Find an eigenvector for each eigenvalue.

Hence find an expression for $x$ in terms of $t$.

Find the time taken for the displacement to fall to 1 cm.

With reference to the eigenvalues, describe the motion of the car body in the long term.

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 $c$.

Let $y=\frac{\mathrm{d}x}{\mathrm{d}t}$. Show that $\frac{\mathrm{d}y}{\mathrm{d}t}=-6x-5y$.

Write down the matrix $\mathit{M}$ such that ${\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}\left(\begin{array}{c}x\\ y\end{array}\right)=\mathit{M}\left(\begin{array}{c}x\\ y\end{array}\right)$.

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

Hence describe the motion of the mass.

By letting $y={\displaystyle \frac{\mathrm{d}x}{\mathrm{d}t}}$, show that the differential equation can be written as $\left(\begin{array}{c}\dot{x}\\ \dot{y}\end{array}\right)=\mathit{M}\left(\begin{array}{c}x\\ y\end{array}\right)$, and write down the matrix $\mathit{M}$.

Use Euler's method with a step length of 0.1 to estimate the angle of the door after 1 second.

Find the eigenvalues of $\mathit{M}$ and a corresponding eigenvector for each.

Hence find the exact solution for $x$ in terms of $t$.

Find the percentage error in your answer to part (b).

State what happens to the door in the long term, and whether it swings past its closed position.

Write the differential equation as a system of two coupled first-order differential equations.

Use Euler's method with a step length of $0.1$ to estimate the displacement of the end of the board when $t=0.5$.