IB Mathematics Applications & Interpretation Topic…

The displacement of a damped spring-mounted platform, $q$ (in $\mathrm{cm}$ ), is modeled by

\frac{d^{2} q}{d t^{2}}+2 \frac{d q}{d t}+5 q=15

(a) Convert the equation to a system of first-order differential equations using $y=\frac{d q}{d t}$ , and find the equilibrium point.

(b) Find the eigenvalues of the homogeneous system's matrix.

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

\frac{\mathrm{d}D}{\mathrm{d}t} = 0.45D - 0.03DW, \quad \frac{\mathrm{d}W}{\mathrm{d}t} = 0.02DW - 0.4W

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

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

(b) 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 $8 \leq W \leq 12$ , the initial direction of the trajectory from $(22,10)$ .

Consider the system of differential equations $\frac{dx}{dt}=ax+2y, \frac{dy}{dt}=2x+ay,$ where $a∈R$ .

(a) Find the eigenvalues of the coefficient matrix and a corresponding eigenvector for each eigenvalue.

Consider the system of differential equations $\frac{dx}{dt}=x+2y, \frac{dy}{dt}=2x+y.$

(a) Find the eigenvalues and a corresponding eigenvector for each eigenvalue. Hence, whether the trajectory along each eigendirection moves towards or away from the origin.

(b) the two eigendirections with arrows. Hence, representative trajectories in the other regions of the phase portrait.

A biologist is studying the interaction between two species in an ecosystem, where the population dynamics can be modeled by the following system of differential equations. Let $x(t)$ and $y(t)$ be the population sizes (in individuals) at time $t$ days.

$\frac{d x}{d t}=2 x+y, \quad \frac{d y}{d t}=4 x+3 y .$

(a) the eigenvalues of the system.

(b) Find the corresponding eigenvectors.

(d) an interpretation of this system in terms of population dynamics.

(d) the graph of $q$ against $t$ for $0 \leq t \leq 5$ , labeling the equilibrium level.

(b) the values of $a$ for which the origin is a saddle point.

(c) Consider the point $(2,1)$ . (i) Express $(\begin{pmatrix}2 \\ 1\end{pmatrix})$ as a linear combination of the two eigenvectors.

(ii) whether the trajectory through $(2,1)$ approaches or moves away from the origin as $t→∞$ . Give a reason for your answer.

IB Mathematics Applications & Interpretation AHL 5.17—phase Portrait Questionbank

IB Mathematics Applications & Interpretation AHL 5.17—phase Portrait Questionbank

IB Mathematics Applications & Interpretation AHL 5.17—phase Portrait Questionbank

IB Mathematics Applications & Interpretation AHL 5.17—phase Portrait Questionbank

IB Mathematics Applications & Interpretation (AI) Topic AHL 5.17—phase Portrait Questionbank