5.18 Second order differential equationsIB Maths: Applications and Interpretation HL: Subtopic test

1. 1
   The displacement $x$ metres of a mass on a spring, $t$ seconds after release, satisfies $\frac{d^2x}{dt^2}=-x$, with $x=1$ and $\frac{dx}{dt}=0$ when $t=0$. Euler's method with step length $h=0.1$ seconds is used, with $y=\frac{dx}{dt}$.

   (a)

   Using $y=\frac{dx}{dt}$, which pair of first-order equations is equivalent to $\frac{d^2x}{dt^2}=-x$?

   [1 mark]

   • A$\frac{dx}{dt}=y$ and $\frac{dy}{dt}=-x$
   • B$\frac{dx}{dt}=-x$ and $\frac{dy}{dt}=y$
   • C$\frac{dx}{dt}=y$ and $\frac{dy}{dt}=x$
   • D$\frac{dx}{dt}=-y$ and $\frac{dy}{dt}=x$

   (b)

   Find the values of $x$ and $y$ when $t=0.1$.

   [1 mark]

   • A$x=0.9$, $y=0$
   • B$x=1$, $y=0.1$
   • C$x=1$, $y=-0.1$
   • D$x=0.9$, $y=-0.1$

   (c)

   Find the approximation to $x$ when $t=0.2$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The displacement $x$ cm of a damped oscillator, $t$ seconds after it starts, satisfies $\frac{d^2x}{dt^2}+2\frac{dx}{dt}+10x=0$. Let $y=\frac{dx}{dt}$.

   (a)

   Which pair of first-order equations is equivalent to the second-order equation?

   [1 mark]

   • A$\frac{dx}{dt}=y$ and $\frac{dy}{dt}=10x+2y$
   • B$\frac{dx}{dt}=y$ and $\frac{dy}{dt}=-10x+2y$
   • C$\frac{dx}{dt}=-2y$ and $\frac{dy}{dt}=-10x$
   • D$\frac{dx}{dt}=y$ and $\frac{dy}{dt}=-10x-2y$

   (b)

   Find the eigenvalues of the matrix that represents the coupled system.

   [1 mark]

   • A$1\pm3i$
   • B$-1\pm3i$
   • C$-2\pm10i$
   • D$-1\pm i\sqrt{10}$

   (c)

   Describe the motion of the oscillator, giving a reason.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A mass on a spring has displacement $x$ metres at time $t$ seconds, where $\frac{d^2x}{dt^2}=-4x-0.5\frac{dx}{dt}$, with $x=2$ and $\frac{dx}{dt}=0$ when $t=0$. Let $y=\frac{dx}{dt}$. Euler's method with step length $h=0.2$ seconds is used.

   (a)

   Write the second-order equation as a pair of coupled first-order equations, and write down the values of $x$ and $y$ when $t=0$.

   [3 marks]

   (b)

   Use Euler's method to find $x$ and $\frac{dx}{dt}$ when $t=0.4$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The angle $\theta$ radians of a pendulum from the vertical, $t$ seconds after release, satisfies $\frac{d^2\theta}{dt^2}=-10\sin\theta-0.5\frac{d\theta}{dt}$, with $\theta=1$ and $\frac{d\theta}{dt}=0$ when $t=0$. Let $\omega=\frac{d\theta}{dt}$. Use radians on your GDC.

   (a)

   (i) Write the equation as a pair of coupled first-order equations.
   (ii) Use Euler's method with step length $h=0.1$ to estimate $\theta$ and $\omega$ when $t=0.2$, giving your answers to 3 significant figures.

   [6 marks]

   (b)

   For small angles, $\sin\theta\approx\theta$.
   (i) Use this to write the equation as a coupled system $\frac{d}{dt}\begin{pmatrix} \theta \\ \omega \end{pmatrix}=M\begin{pmatrix} \theta \\ \omega \end{pmatrix}$, and show that the eigenvalues of $M$ satisfy $\lambda^2+0.5\lambda+10=0$.
   (ii) Find the eigenvalues of $M$.
   (iii) Describe the long-term motion of the pendulum, giving a reason.

   [6 marks]

   Total for question 4: 12 marks

End of questions