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）
Exercise 1.
(1) Solve the following integral equation. $y(x)=e^x\left(1+\int_0^x e^{-t} y(t) d t\right)$
(2) Show that the differential equation
$$
y^{\prime \prime}+a^2 y=f(x), y(0)=y^{\prime}(0)=0
$$
has $y(x)=\frac{1}{a} \int_0^x f(t) \sin a(x-t) d t$ as its solution. Hint: in both cases use the Laplace transform and the convolution theorem.
Exercise 2.
Find the convolution of each of the following pairs of functions:
(1) $t$ and $e^{a t}$
(2) $e^{a t}$ and $\cos (b t)$
(3) $e^{a t}$ and $e^{b t}$
Exercise 3.
(1) Solve the following initial value problem: $y^{\prime \prime}+2 y^{\prime}+y=f(x), y(0)=y^{\prime}(0)=0$, where
$$
f(x)= \begin{cases}e^{-x} & \text { if } 0 \leq x<4 \ 0 & \text { if } 4 \leq x\end{cases}
$$
(2) Solve the following initial value problem: $y^{\prime \prime}-2 y^{\prime}-y=\delta, y(0)=y^{\prime}(0)=0$, where $\delta$ is the Dirac delta function.
Exercise 4.
Suppose a tank holds a brine solution consisting of $2 \mathrm{~kg}$ salt dissolved in $10 \mathrm{~L}$ of water. There are two input sources of brine solution: the first source has a concentration of $1 \mathrm{~kg}$ of salt per liter while the second source is pure water. The first source flows into the tank at a rate of $3 \mathrm{~L} / \mathrm{min}$ for $2 \mathrm{~min}$. Thereafter, it is turned off and simultaneously the second source is turned on at a rate of $3 \mathrm{~L} / \mathrm{min}$ for $2 \mathrm{~min}$. Thereafter, it is turned off and simultaneously the first source is turned back on at a rate of 3 $\mathrm{L} / \mathrm{min}$ and remains on. The well-mixed solution flows out of the $\operatorname{tank}$ at a rate of $3 \mathrm{~L} / \mathrm{min}$. Find the amount of salt in the tank at time $t$.
Exercise 5.
(1) Let $n, m$ be positive integers. Use the Convolution Theorem to evaluate the integral $\int_0^1 s^m(1-s)^n d s$
(2) Verify the following properties of convolution:
$(f+g) * h=f * h+g * h$
$(f * g) * h=f *(g * h)$
(3) It is tempting to guess that the convolution operation satisfies some of the other (natural) properties of the standard product, such as $f \cdot f \geq 0$ or $f \cdot 1=f$. Here, you are asked to show that this is not the case.
Give an example of a continuous function $f$ which satisfies that $f * 1 \neq f$.
Give an example of a continuous function $g$ for which $(g * g)(x)<0$ for some $x$.
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！
