IB Mathematics Applications & Interpretation AHL 5.16—eulers Method for 1st Order Des Questionbank

(e) The analytical solution is $v=7\tanh(1.4t)$. Use it to find the actual velocity at $t=0.2$.

A storage tank is being filled, and the water depth $d(t)$ metres after $t$ minutes is modelled by $\frac{dd}{dt}=1.2-0.08d^2$, with $d(0)=2$.

(a) Using Euler's method with a step size $h=0.5$, approximate $d(1)$.

(c) Explain one disadvantage of using a very small step size in Euler's method.

An object is released from rest at the top of a long vertical tube filled with a thick viscous fluid at time $t = 0$ seconds. Take $x(0)=0$ and $v(0)=0$, where $x$ is the object’s displacement from the top of the tube measured in metres, and $v=\frac{\text{d}x}{\text{d}t}$ is its velocity in $\text{m s}^{-1}$.

Initially, the resistance is modelled as proportional to the velocity, giving the differential equation

$\frac{\text{d}^2x}{\text{d}t^2} = 9.81 - 0.9\left( \frac{\text{d}x}{\text{d}t} \right)$

where $9.81$ is in $\text{m s}^{-2}$ and the constant $0.9$ has units $\text{s}^{-1}$.

The maximum velocity approached by the object as it falls is known as the terminal velocity.

An experiment is performed in which the object is placed in the fluid on a number of occasions and its terminal velocity is recorded. It is found that the terminal velocity was consistently smaller than that predicted by the model used. It was suggested that the resistance to motion is actually proportional to the velocity squared and so the following model was set up:

$\frac{\text{d}^2x}{\text{d}t^2} = 9.81 - 0.9\left( \frac{\text{d}x}{\text{d}t} \right)^2$

In this second model, the constant $0.9$ has units $\text{m}^{-1}$.

At terminal velocity, the acceleration of an object is equal to zero.

(a) By substituting $v = \frac{\text{d}x}{\text{d}t}$ into the equation, find an expression for the velocity of the object at time $t$. Give your answer in the form $v = f(t)$.

(b) From your solution to part (a), or otherwise, find the terminal velocity of the object predicted by this model.

(c) Write down the differential equation for the second model as a system of first-order differential equations.

(d) Use Euler’s method, with a step length of $0.2\,\text{s}$, to find the displacement and velocity of the object when $t = 0.6$.

A predator-prey system models the populations of deer ($D$, in thousands) and wolves ($W$, in hundreds) in a national park, given by:

where $t$ is in years. Initially, $D = 22, W = 10$.

(a) Find the non-zero equilibrium point for the system.

(b) Interpret the equilibrium point in the context of the model.

(d) On a phase plane with $D$ on the horizontal axis and $W$ on the vertical axis, for $18 \leq D \leq 24$ and $8 \leq W \leq 12$, sketch the initial direction of the trajectory from $(22,10)$.

A tank contains a saline solution with concentration $S$ (in $\mathrm{g\,L^{-1}}$) at time $t$ (in minutes). Water with salt at $2\mathrm{~g\,L^{-1}}$ flows in at $3\mathrm{~L\,min^{-1}}$, and the solution flows out at $3\mathrm{~L\,min^{-1}}$. The tank volume is $100\mathrm{~L}$. The differential equation is:

$\frac{\mathrm{d}S}{\mathrm{d}t} = 0.06 - 0.03S$

(Here, $\frac{\mathrm{d}S}{\mathrm{d}t}$ is in $\mathrm{g\,L^{-1}\,min^{-1}}$, is in , and is in .)

The initial concentration is $S = 1\mathrm{~g\,L^{-1}}$ at $t = 0$.

(a) Solve the differential equation to find $S(t)$.

(d) Sketch the solution curve for $0\le t\le 2$, indicating the Euler method points and equilibrium.

(b) Comment on whether using a smaller step size, such as $h=0.25$, would lead to a more accurate solution.

(c) Use Euler's method with a step size of $0.2$ years to estimate $D$ and $W$ after $0.2$ years.

(b) Use Euler's method with $h = 1$ to estimate $S(2)$.

(c) Sketch the analytical solution curve $S(t)$ and the Euler approximations on the same axes for $0 \leq t \leq 5$.

(d) Determine the time when $S = 1.5\mathrm{~g\,L^{-1}}$ using the analytical solution.