### 数学代写|常微分方程代写ordinary differential equation代考|MATH53

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• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|常微分方程代写ordinary differential equation代考|Ordinary differential equations

The term ‘equatio differentialis’ (differential equations) was first used by Leibniz in 1676 to denote a relationship between the differentials of two variables. Very soon, this restricted usage was abandoned. Roughly speaking, differential equations are the equations involving one or more dependent variables (unknowns) and their derivatives/partial derivatives. If the unknown in the differential equation is a function of only one variable, then such differential equation is called an ordinary differential equation (ODE).
Notation: Unless specified otherwise, the unknown in the differential equation is denoted by $y$. Let $\mathbb{R}$ denote the set of real numbers, and $J$ be an open interval in $\mathbb{R}$. Throughout the book we denote the derivative of the function $y: J \rightarrow \mathbb{R}$ with respect to $x$ by either
$$\frac{d}{d x} y(x) \text { or } \frac{d y}{d x}(x) \text { or } y^{\prime}(x) .$$
When there is no ambiguity regarding the argument in the function $y$, we denote the derivative simply with $\frac{d y}{d x}$ or $y^{\prime}$. Similarly, let $y^{\prime \prime}$ and $y^{\prime \prime \prime}$ denote the second and the third derivative of $y$, respectively. In general, for $k \in \mathbb{N}$, $y^{(k)}$ or $\frac{d^k y}{d x^k}$ denotes the $k$-th order derivative of $y$.
With this notation, examples of ODEs are
$$\begin{gathered} \frac{d}{d x} y(x)=\left(\frac{d^2}{d x^2} y(x)\right)^5+y^2(x), x \in(0,1), \ y^{\prime}=3 y^2+(\sin x) y+\log \left(\cos ^2 y\right), x \in \mathbb{R} . \end{gathered}$$
The order of an ODE is the largest number $k$ such that the $k$-th order derivative of the unknown is present in the ODE. For example, the order of (1.1) is two.
At the beginning, it may look like tools from the integral calculus are sufficient to study ODEs. But very soon one realizes that to develop methods to solve or analyze them, one needs notions from subjects like analysis, linear algebra, etc. In fact, the study of differential equations motivated crucial development of many areas of mathematics: the theory of Fourier series and more general orthogonal expansions, integral transformations, Hilbert spaces, and Lebesgue integration to name a few.

## 数学代写|常微分方程代写ordinary differential equation代考|Applications of ODEs

Many laws in physics, chemistry, biology etc., can be easily expressed using differential equations. One of the reasons for this is the following. The quantity $y^{\prime}(x)$ can be interpreted as the rate of change of the quantity $y$ with respect to the quantity $x$. In many natural phenomena, there is a relationship between the unknowns (which are relatively difficult to measure), the rate of change of the unknowns with respect to a known quantity, and the other known quantities (which are easy to measure) that govern the process. When this relationship is expressed in mathematics, it turns out to be a (system of) differential equation(s). Therefore the study of ODEs is crucial in understanding physical sciences. In fact, much of the theory developed in ODEs owes to the questions/situations raised in the study of subjects like mechanics, astronomy, electronics etc.
Listing all the available ODE models in any branch of science is an impossible task. Therefore in this chapter, we present a few ODE models which arise from physics and biology which can be solved or analyzed using the material in the book. We begin with models from physics.

Example 1.2.1 (Radioactivity and half-life). Let $N(t)$ denote the number of radioactive active atoms in a substance of a fixed quantity at time $t$. Then a model for the decay of the number of radioactive atoms is
$$\begin{gathered} \frac{d}{d t} N(t)=-k N(t), t>0, \ N\left(t_0\right)=N_0, \end{gathered}$$
where $k>0$. Equation (1.3b) is known as the initial condition. This kind of models are studied in detail in Chapter 2, Subsection 2.1.3. One can easily verify that the solution to (1.3a) is
$$N(t)=N_0 e^{-k\left(t-t_0\right)}, t>t_0 .$$
The half-life of a specific radioactive isotope is defined as the time taken for half of its radioactive atoms to decay. In fact, the half-life is independent of the quantity of the radioactive material. We now calculate the half-life of an isotope using (1.3a) if $k$ is known explicitly. For, it is enough to find $T$ at which $N(T)=\frac{N_0}{2}$. From (1.4) we have
$$N(T)=N_0 e^{-k\left(T-t_0\right)}=\frac{N_0}{2}$$

# 常微分方程代写

## 数学代写|常微分方程代写ordinary differential equation代考|Ordinary differential equations

$$\frac{d}{d x} y(x) \text { or } \frac{d y}{d x}(x) \text { or } y^{\prime}(x) .$$

$$\frac{d}{d x} y(x)=\left(\frac{d^2}{d x^2} y(x)\right)^5+y^2(x), x \in(0,1), y^{\prime}=3 y^2+(\sin x) y+\log \left(\cos ^2 y\right), x \in \mathbb{R}$$
$\mathrm{ODE}$ 的阶数是最大数 $k$ 这样的 $k \mathrm{ODE}$ 中存在末知数的 -th 阶导数。例如，(1.1) 的阶数为二。

## 数学代写|常微分方程代写ordinary differential equation代考|Applications of ODEs

$$\frac{d}{d t} N(t)=-k N(t), t>0, N\left(t_0\right)=N_0,$$

$$N(t)=N_0 e^{-k\left(t-t_0\right)}, t>t_0 .$$

$$N(T)=N_0 e^{-k\left(T-t_0\right)}=\frac{N_0}{2}$$

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