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）
Solve each of the following differential equations by the method of Laplace transform:
(1) $y^{\prime \prime}-4 y^{\prime}+4 y=0, y(0)=0$ and $y^{\prime}(0)=3$
(2) $y^{\prime \prime}+y^{\prime}=3 x^2, y(0)=0$ and $y^{\prime}(0)=1$
(3) $x y^{\prime \prime}+(3 x-1) y^{\prime}-(4 x+9) y=0, y(0)=0$
(4) $y^{\prime \prime}-6 y^{\prime}+9 y=(x+1) e^{3 x}$. Is it enough to give the condition $y(0)=y_0$ to find the unique solution?
(5) $y^{\prime \prime \prime}-6 y^{\prime \prime}+13 y^{\prime}-10 y=e^{2 x} \cos (x)$. Determine a sufficient number of initial conditions which allow you to find a unique solution!
Find the inverse Laplace transforms of the following functions:
(1) $\frac{p+3}{p^2+2 p+5}$
(2) $\frac{1}{p^3}+\frac{8}{p^2+4}$
(3) $\frac{1}{\left(p^2+a^2\right)^2}$, where $a \in \mathbb{R}$ and $p$ is the standard variable used when defining the Laplace transform.
(4) $\frac{1}{(p+9)^4}$.
In each of the following cases, graph the function and find its Laplace transform:
(1) $f(x)=u(x-a)$ where $a$ is a positive number and $u(x)$ is the unit step function defined by
$$
u(x)= \begin{cases}0 & \text { if } x<0 \ 1 & \text { if } x \geq 0\end{cases}
$$
(2) $f(x)=[x]$ where $[x]$ denotes the greatest integer $\leq x$
(3) $f(x)=x-[x]$
Recall that a function $f:[0,+\infty) \rightarrow \mathbb{R}$ is of exponential order if there are constants $C, a$ such that $|f(x)| \leq C e^{a x}$ when $x \rightarrow+\infty$.
(1) Prove that the function $e^{x^2}$ is not of exponential order.
(2) Show that there is no $p$ for which $\mathscr{L}\lefte^{x^2}\right$ exists.
Compute $\int_0^{+\infty} \frac{\cos (a t)}{1+t^2} \mathrm{~d} t$, where $a \in \mathbb{R}$. HINT: a possible way to compute this is to regard this integral as $f(a)$, i.e. a function of $a$, and compute its Laplace transform $\mathscr{L}(f)$. Interchanging the two integrals, you might find some well-known formulas.
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！
