数学代写|MATH1101 Mathematical Modeling

Consider the linear system $\mathbf{x}^{\prime}(t)=\mathbf{A} \mathbf{x}=\left(\begin{array}{ll}1 & \alpha \ 1 & 1\end{array}\right) \mathbf{x}$, where $\alpha$ is a constant.
a. Show that the eigenvalues of this system are $\lambda=1 \pm \sqrt{\alpha}$.
b. Explain how the orbits of the system $\mathbf{x}^{\prime}(t)=\mathbf{A x}$ change as $\alpha$ passes through zero from negative to positive, but remains less than 1 ..
c. Illustrate your answer in part b. by plotting phase portraits of the given system for $\alpha=-0.49, \alpha=0$, and $\alpha=0.49$.
d. Repeat part b. for the transition through $\alpha=1$. Illustrate your answer by plotting phase portraits of the given system for $\alpha=0.81, \alpha=1$, and $\alpha=1.44$. [Hint: Use a symmetric window, say $-1 \leq x \leq 1$ and $-1 \leq y \leq 1$ for the portraits in parts c. and d.; you will likely need to vary the initial conditions for different values of $\alpha$ in order to get ‘nice’ phase portraits for each one.]

Consider the constant coefficient second order linear DE
$$
a y^{\prime \prime}+b y^{\prime}+c y=0 .
$$
a. Assuming $a \neq 0$, write down the equivalent dynamical system for the given DE. Show that the eigenvalues of this system are equivalent to the roots of the characteristic equation $Z(\lambda)=0$ of the given DE.
b. For each of the DEs i., ii., and iii.:
find the characteristic roots and state the general solution $y(t)$;
use the fact that the characteristic roots are the eigenvalues to classify the equivalent dynamical system as to type and stability, and state the vector solution $\mathbf{x}(t)=\left(x_1(t), x_2(t)\right)^T=\left(y(t), y^{\prime}(t)\right)^T$;
sketch the phase portrait (use Maple if desired).
i. $2 y^{\prime \prime}-3 y^{\prime}+y=0$
ii. $y^{\prime \prime}-2 y^{\prime}+2 y=0$
iii. $y^{\prime \prime}+5 y^{\prime}=0$

A mass $m=0.1 \mathrm{~kg}$ stretches a spring $0.05 \mathrm{~m}$ when hanging at rest.
a. Find the value of the spring constant $k$.
b. If the mass is set in motion from equilibrium $y_0=0$ with a downward velocity $v_0=0.1 \mathrm{~m} \mathrm{~s}^{-1}$, find the position $y(t)$ of the mass at any time $t \geq 0$. State the amplitude, frequency, and period of the motion.
c. What is the time $t^{\star}$ when the mass first returns to equilibrium?
d. Suppose the above mass moves in a resistive medium which exerts a drag force of $\gamma v$, where $\gamma=8 \mathrm{~kg} \mathrm{~s}^{-1}$. Will the mass still oscillate, i.e., be underdamped? Explain.

Consider the standard mass-spring-damper model
$$
m y^{\prime \prime}+\gamma y^{\prime}+k y=0 .
$$
a Show that, with the choice of dimensionless variable $\tau=\omega_0 t=\sqrt{\frac{k}{m}} t$, this DE becomes
$$
\frac{d^2 y}{d \tau^2}+2 \zeta \frac{d y}{d \tau}+y=0, \quad(\star)
$$
where $\zeta=\frac{\gamma}{2 \sqrt{k m}}$.

b. Prove that the characteristic roots of the DE $(\star)$ are either negative reals, or have a negative real part. (Thus such motion always dies off as $t \rightarrow \infty$, and the corresponding dynamical system is always asymptotically stable.)