MY-ASSIGNMENTEXPERT™可以为您提供sydney MATH3063 Ordinary Differential Equations常微分方程的代写代考和辅导服务！
这是悉尼大学 常微分方程课程的代写成功案例。
MATH3063课程简介
This unit of study is an introduction to the theory of systems of ordinary differential equations. Such systems model many types of phenomena in engineering, biology and the physical sciences. The emphasis will not be on finding explicit solutions, but instead on the qualitative features of these systems, such as stability, instability and oscillatory behaviour. The aim is to develop a good geometrical intuition into the behaviour of solutions to such systems. Some background in linear algebra, and familiarity with concepts such as limits and continuity, will be assumed. The applications in this unit are from biology, physics, chemistry, and engineering, including population dynamics, epidemics, chemical reactions, and simple mechanical systems.
Prerequisites
At the completion of this unit, you should be able to:
- LO1. explain the principle of linear approximations to nonlinear systems and use this to analyse system behaviour close to steady states
- LO2. synthesise graphical information from nullclines and flow to construct qualitative phase plane solutions to problems in nonlinear systems
- LO3. demonstrate knowledge of the theory of existence and uniqueness, and the determination of stability of solutions of ordinary differential equations, including special cases such as Hamiltonian and gradient systems.
- LO4. interpret model results and evaluate and explain the limitations of models in representing real systems
- LO5. demonstrate a broad understanding of the role of basic bifurcations in nonlinear systems and evaluate the effect of parameter variation on observed model behaviour
- LO6. apply mathematical theory in novel and diverse applications.
MATH3063 Ordinary Differential Equations HELP（EXAM HELP， ONLINE TUTOR）
Consider the initial value problem
$$
y^{\prime}=y^{\frac{1}{4}}, \quad y(0)=0
$$
(1) Show that the problem fails to satisfy the assumptions of the local existence and uniqueness theorem (Theorem A in the textbook).
(2) Show that for every value $h>0$ the problem has at least two distinct solutions $y_1, y_2$ which are defined on the interval $x \in[-h, h]$.
Use Picard’s existence and uniqueness results to show that if $[a, b] \subset \mathbb{R}$ is a closed interval, $q \in \mathbb{R}$, $f$ is a continuous function on $[a, b]$, and $x_0 \in(a, b)$, then there is a unique solution to the initial value problem
$$
y^{\prime}+q y=f(x), \quad y\left(x_0\right)=y_0
$$
on some interval $I$ whose interior contains $x_0$.
We are interested in the initial value problems
$$
\left{\begin{array}{l}
y^{\prime}=e^{\alpha x} e^{\beta y}=: f(x, y), \
y\left(x_0\right)=y_0 .
\end{array}\right.
$$
(1) Show that $f$ is locally Lipschitz continuous with respect to the $y$ variable. Show that if $\beta \geq 0$, then it is Lipschitz continuous on $(-\infty, d]$ for any $d \in \mathbb{R}$ and compute its Lipschitz constant in this case.
(2) Show that for any $\beta \neq 0, f$ is not Lipschitz continuous w.r.t. the y variable on the whole $\mathbb{R}$.
(3) What can we say about the existence and uniqueness of solutions to the previous problem, when $\beta \geq 0$ and $\alpha \leq 0$ ?. Justify your answer! Hint: here you just need to rely on Picards theorems.
(4) In the case of (3), can we have solutions that are defined on the whole $\mathbb{R}$ ? Justify your answer.
(5) Take $\alpha=-1$ and $\beta=2$. Find an initial condition $\left(x_0, y_0\right)$ for which the solution of the previous ODE does not exist for all $x \geq 0$.
(6) In the case of (5), is it possible to find initial conditions $\left(x_0, y_0\right)$ such that the solution of the ODE exists for all $x \leq 0$ ?
(7) In the case of (5), can one find initial conditions such that the solution of the ODE exists on some interval $(a, b)$, but $b$ cannot be arbitrary large and $a$ cannot be arbitrary small in the same time?
(1) Construct a Lipschitz function on the whole real line, which is not differentiable at exactly 3 points.
(2) Show that any continuous piecewise affine function is globally Lipschitz continuous. Compute also its Lipschitz constant in this case. A piecewise affine function on $\mathbb{R}$ can be defined as follows: consider $\left(I_i\right){i=1}^n$ a finite partition of $\mathbb{R}$, i.e. $n \in \mathbb{N}$ is finite and $I_i \cap I_j=\emptyset$ if $i \neq j$ and $\cup{i=1}^n I_n=\mathbb{R}$. Then we consider $\left(a_i\right){i=1}^n$ and $\left(b_i\right){i=1}^n$ real numbers and define $f: \mathbb{R} \rightarrow \mathbb{R}$ as
$$
f(x)=a_i x+b_i, \text { if } x \in I_i .
$$
In addition, since $f$ is supposed to be continuous, the coefficients $\left(a_i, b_i\right)$ are chosen accordingly.
Prove the following identities for every $L>0$ and integers $n, m \geq 1$
(1) $\int_{-L}^L \cos \left(\frac{\pi}{L} n x\right) \cos \left(\frac{\pi}{L} m x\right) d x= \begin{cases}0 & \text { if } n \neq m \ L & \text { if } n=m\end{cases}$
(2) $\int_{-L}^L \sin \left(\frac{\pi}{L} n x\right) \sin \left(\frac{\pi}{L} m x\right) d x= \begin{cases}0 & \text { if } n \neq m \ L & \text { if } n=m\end{cases}$
(2) $\int_{-L}^L \cos \left(\frac{\pi}{L} n x\right) \sin \left(\frac{\pi}{L} m x\right) d x=0$
MY-ASSIGNMENTEXPERT™可以为您提供SYDNEY MATH3063 ORDINARY DIFFERENTIAL EQUATIONS常微分方程的代写代考和辅导服务！
