IB Mathematics Applications & Interpretation Topic…

The motion of a particle is given by:

\ddot{x}+5 \dot{x}^{2}=2 \sin (t),

with $x$ in metres, $t$ in seconds $(t\geq 0)$ , and $\sin(t)$ using radian measure.

Convert to a system of first-order differential equations.

With $x(0)=1, \dot{x}(0)=-1$ , use Euler's method with step size 0.2 to and at .

A particle moves along a straight line. Its displacement $x$ (in metres) at time $t$ (in seconds), $t\geq 0$ , satisfies

A spring system is governed by:

\ddot{x}+3 \dot{x}+2 x=\cos (2 t),

A system is modeled by:

\ddot{x}+2 \dot{x}+3 x=t^{3},

A particle's motion is governed by the second-order differential equation

\ddot{x}+3 \dot{x}+2 x=\sin (t),

Hence the acceleration at $t=0.6$ .

one limitation of Euler's method in this context.

Express this as a system of first-order differential equations.

With $x(0)=0, \dot{x}(0)=1$ , use Euler's method with step size $0.2$ to $x$ and $\dot{x}$ at $t=0.4$ .

Use your estimate of $\dot{x}(0.4)$ from Part 1 to the acceleration at $t=0.4$ .

why Euler's method might overestimate or underestimate the velocity at $t=0.4$ .

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

Given $x(0)=0$ and $\dot{x}(0)=0$ , use Euler's method with step size $0.1$ to $x$ and $\dot{x}$ at $t=0.5$ .

Hence, the graph of $\dot{x}$ against $t$ for $0 \le t \le 0.5$ , using the Euler estimates from the previous part.

Write as a system of first-order differential equations.

With $x(0)=0, \dot{x}(0)=0$ , use Euler's method with step size $0.1\,\text{s}$ to $x$ and $\dot{x}$ at $t=0.3\,\text{s}$ .

Using Euler estimates at $t=0,0.1,0.2,0.3$ , the displacement $x$ against $t$ for $0\leq t\leq 0.3$ .

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

Given initial conditions $x(0)=0, \dot{x}(0)=1$ , use Euler's method with a step size of $0.1$ to the displacement and velocity at $t=0.5$ .

the approximate solution for displacement $x$ from $t=0$ to $t=0.5$ using the points calculated in Part 1.

IB Mathematics Applications & Interpretation AHL 5.18—eulers Method for 2nd Order Des Questionbank