5.18 Second order differential equationsIB Maths: Applications and Interpretation HL: Flashcards

What these 14 flashcards ask

How do you write \frac{d^2x}{dt^2}=f(x,y,t) as a coupled first-order system?
What is the initial value of y in the coupled system?
Euler's method for the coupled pair \frac{dx}{dt}=y, \frac{dy}{dt}=f(x,y,t)?
Which values are used on the right in each Euler step?
Write \frac{d^2x}{dt^2}=-9x as a coupled system.
Matrix for \frac{d^2x}{dt^2}+a\frac{dx}{dt}+bx=0 with y=\frac{dx}{dt}?
Eigenvalue equation for \frac{d^2x}{dt^2}+a\frac{dx}{dt}+bx=0?
\frac{d^2x}{dt^2}+4x=0: what are the eigenvalues and the phase portrait?
What does a spiral towards the origin mean for the motion?
What do complex eigenvalues tell you about x(t)?
What do two negative real eigenvalues tell you about x(t)?
What do real eigenvalues of opposite signs mean for a spring equation?
What do the axes of the phase portrait show?
Why must you use radians on the GDC for a pendulum equation?

Exam questions on 5.18 Second order differential equations

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 $y=\frac{dx}{dt}$ .
Find the approximation to $x$ when $t=0.2$ .2 marks
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 $y=\frac{dx}{dt}$ .
Describe the motion of the oscillator, giving a reason.2 marks
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.
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

Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).