5.18 Second order differential equationsIB Maths: Applications and Interpretation HL: Mind map

Exam questions on 5.18 Second order differential equations

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