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

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

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 write the differential equation as a system of coupled, first order differential equations.

Use Euler’s method with a step length of $0.1$ to find the displacement of the particle when $t=1$.

Find the long-term velocity of the particle.

$y=\frac{dx}{dt}\Rightarrow \frac{dy}{dt}+5\frac{dx}{dt}+6x=0$   OR   $\frac{dy}{dt}+5y+6x=0$         M1

Note: Award M1 for substituting $\frac{dy}{dt}$ for $\frac{d^{2}x}{d{t}^2}$.

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

[1 mark]

$\text{det}\left(\begin{array}{cc}-\lambda & 1\\ -6 & -5-\lambda \end{array}\right)=0$         (M1)

Note: Award M1 for an attempt to find eigenvalues. Any indication that $\text{det}\left(\mathit{M}-\lambda \mathit{I}\right)=0$ has been used is sufficient for the (M1).

$-\lambda \left(-5-\lambda \right)+6=0$   OR   ${\lambda }^2+5\lambda +6=0$         (A1)

$\lambda =-2, -3$        A1

[3 marks]

(on a phase portrait the particle approaches $(0, 0)$ as $t$ increases so long term velocity ($y$) is)

$0$        A1

Note: Only award A1 for $0$ if both eigenvalues in part (a)(ii) are negative. If at least one is positive accept an answer of ‘no limit’ or ‘infinity’, or in the case of one positive and one negative also accept ‘no limit or $\mathit{0}$ (depending on initial conditions)’.

[1 mark]

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

$\frac{d^{2}x}{d{t}^2}=\frac{dy}{dt}$         (A1)

$\frac{dy}{dt}+5y+6x=3t+4$        A1

[2 marks]

recognition that $h=0.1$ in any recurrence formula           (M1)

$\left(t_{n+1}=t_n+0.1\right)$

$x_{n+1}=x_n+0.1y_n$           (A1)

$y_{n+1}=y_n+0.1\left(3t_n+4-5y_n-6x_n\right)$           (A1)

(when $t=1$,) $x=0.64402\dots \approx 0.644 \text{m}$        A2

[5 marks]

recognizing that $y$ is the velocity

$0.5 {\text{m\hspace{0.05em}s}}^{-1}$         A1 

[2 marks]

It was clear that second order differential equations had not been covered by many schools. Fortunately, many were able to successfully answer part (ii) as this was independent of the other two parts. For part (iii) it was expected that candidates would know that two negative eigenvalues mean the system tends to the origin and so the long-term velocity is 0. Some candidates tried to solve the system. It should be noted that when the command term is ‘state’ then no further working out is expected to be seen.

Forming a coupled system from a second order differential equation and solving it using Euler’s method is a technique included in the course guide. Candidates who had learned this technique were successful in this question.