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## 数学代写|微积分代写Calculus代考|Differential Equations-Growth and Decay

Topics

• Verifying solutions to differential equations.
• Separation of variables.
• Euler’s method.
• Growth and decay models.
• Applications.
Definitions and Theorems
• A differential equation in $x$ and $y$ is an equation that involves $x, y$, and derivatives of $y$.
• A function $f(x)$ is a solution to a differential equation if the equation is satisfied when $y$ and its derivatives are replaced by $f(x)$ and its derivatives.
• Euler’s method for approximating the solution to $y^{\prime}=F(x, y), y\left(x_0\right)=y_0$ is given by $x_n=x_{n-1}+h, y_n=y_{n-1}+h F\left(x_{n-1}, y_{n-1}\right)$. Here, $F(x, y)$ is a convenient notation to indicate that $y^{\prime}$ equals an expression involving both $x$ and $y$.
• The solution to the growth and decay model $\frac{d y}{d t}=k y$ is $y=C e^{k t}$.

## 数学代写|微积分代写Calculus代考|Study Tips

Study Tips

• The general solution to a differential equation will contain one or more arbitrary constants. You need to have appropriate initial conditions to determine these constants.
• You can always check your solution to a differential equation by substituting it back into the original equation.
• Euler’s method is the simplest of all the numerical methods for solving differential equations. You can obtain more accurate approximations by using a smaller step size, or a more accurate method, such as the Runge-Kutta method.
• For the growth and decay model $y=C e^{k t}$, there is growth when $k>0$ and decay when $k<0$.
• Logarithmic manipulations play an important role in solving growth and decay applications. For example, the following is how to solve for an exponent $k$.
$$e^{k(5715)}=1 / 2 \Rightarrow \ln e^{k(5715)}=\ln 1 / 2 \Rightarrow k(5715)=\ln 1 / 2 \text {. Finally, } k=\frac{\ln 1 / 2}{5715}=\frac{-\ln 2}{5715} .$$
Pitfalls
• Be careful when working with logarithms. For example, in the computation $-2 k=\ln (3 / 7) \Rightarrow k \approx 0.4236$, the final answer is positive because $\ln (3 / 7)<0$.
• Unfortunately, not all differential equations can be solved by separation of variables. To use this method, you have to be able to move all of the terms containing $x$ to one side and all of the terms containing $y$ to the other side.

## 数学代写|微积分代写微积分代考|微分方程-生长与衰减

. . .

Topics

• 微分方程解的验证。
• 分离变量。
• 欧拉方法。
• 生长衰减模型。
• 应用。
定义与定理
• 中的一个微分方程 $x$ 和 $y$ 方程是否包含 $x, y$的导数 $y$.
• 一个函数 $f(x)$ 当方程满足时，微分方程是否有解 $y$ 它的导数被 $f(x)$ 和它的导数。
• 近似解的欧拉方法 $y^{\prime}=F(x, y), y\left(x_0\right)=y_0$ 由 $x_n=x_{n-1}+h, y_n=y_{n-1}+h F\left(x_{n-1}, y_{n-1}\right)$。这里， $F(x, y)$ 有方便的符号来表示吗 $y^{\prime}$ 等于包含两者的表达式 $x$ 和 $y$.
• 生长衰减模型的解 $\frac{d y}{d t}=k y$ 是 $y=C e^{k t}$.

## 数学代写|微积分代写Calculus代考|Study Tips

.

$$e^{k(5715)}=1 / 2 \Rightarrow \ln e^{k(5715)}=\ln 1 / 2 \Rightarrow k(5715)=\ln 1 / 2 \text {. Finally, } k=\frac{\ln 1 / 2}{5715}=\frac{-\ln 2}{5715} .$$

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

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## 数学代考|线性代数代写LINEAR ALGEBRA代考|Application to Engin

A number of important engineering problems, particularly in electrical engineering and control theory, can be analyzed by Laplace transforms. This approach converts an appropriate system of linear differential equations into a system of linear algebraic equations whose coefficients involve a parameter. The next example illustrates the type of algebraic system that may arise.

EXAMPLE 2 Consider the following system in which $s$ is an unspecified parameter. Determine the values of $s$ for which the system has a unique solution, and use Cramer’s rule to describe the solution.
$$\begin{array}{r} 3 s x_{1}-2 x_{2}=4 \ -6 x_{1}+s x_{2}=1 \end{array}$$
SOLUTION View the system as $A \mathbf{x}=\mathbf{b}$. Then
$$A=\left[\begin{array}{cr} 3 s & -2 \ -6 & s \end{array}\right], \quad A_{1}(\mathbf{b})=\left[\begin{array}{cc} 4 & -2 \ 1 & s \end{array}\right], \quad A_{2}(\mathbf{b})=\left[\begin{array}{cc} 3 s & 4 \ -6 & 1 \end{array}\right]$$
Since
$$\operatorname{det} A=3 s^{2}-12=3(s+2)(s-2)$$
the system has a unique solution precisely when $s \neq \pm 2$. For such an $s$, the solution is $\left(x_{1}, x_{2}\right)$, where
\begin{aligned} &x_{1}=\frac{\operatorname{det} A_{1}(\mathbf{b})}{\operatorname{det} A}=\frac{4 s+2}{3(s+2)(s-2)} \ &x_{2}=\frac{\operatorname{det} A_{2}(\mathbf{b})}{\operatorname{det} A}=\frac{3 s+24}{3(s+2)(s-2)}=\frac{s+8}{(s+2)(s-2)} \end{aligned}

## 数学代考|线性代数代写LINEAR ALGEBRA代考|A Formula for A–1

Cramer’s rule leads easily to a general formula for the inverse of an $n \times n$ matrix $A$. The $j$ th column of $A^{-1}$ is a vector $\mathbf{x}$ that satisfies
$$A \mathbf{x}=\mathbf{e}{j}$$ where $\mathbf{e}{j}$ is the $j$ th column of the identity matrix, and the $i$ th entry of $\mathbf{x}$ is the (i,j)-entry of $A^{-1}$. By Cramer’s rule,
$$\left{(i, j) \text {-entry of } A^{-1}\right}=x_{i}=\frac{\operatorname{det} A_{i}\left(\mathbf{e}{j}\right)}{\operatorname{det} A}$$ Recall that $A{j i}$ denotes the submatrix of $A$ formed by deleting row $j$ and column $i$. A cofactor expansion down column $i$ of $A_{i}\left(\mathbf{e}{j}\right)$ shows that $$\operatorname{det} A{i}\left(\mathbf{e}{j}\right)=(-1)^{i+j} \operatorname{det} A{j i}=C_{j i}$$
where $C_{j i}$ is a cofactor of $A$. By (2), the (i,j)-entry of $A^{-1}$ is the cofactor $C_{j i}$ divided by $\operatorname{det} A$. [Note that the subscripts on $C_{j i}$ are the reverse of $(i, j)$.] Thus
$$A^{-1}=\frac{1}{\operatorname{det} A}\left[\begin{array}{rrrr} C_{11} & C_{21} & \cdots & C_{n 1} \ C_{12} & C_{22} & \cdots & C_{n 2} \ \vdots & \vdots & & \vdots \ C_{1 n} & C_{2 n} & \cdots & C_{n n} \end{array}\right]$$
The matrix of cofactors on the right side of (4) is called the adjugate (or classical adjoint) of $A$, denoted by adj $A$. (The term adjoint also has another meaning in advanced texts on linear transformations.) The next theorem simply restates (4).

## 数学代考|线性代数代写LINEAR ALGEBRA代考|Application to Engin

$$3 s x_{1}-2 x_{2}=4-6 x_{1}+s x_{2}=1$$

$$A=\left[\begin{array}{lll} 3 s & -2-6 & s \end{array}\right], \quad A_{1}(\mathbf{b})=\left[\begin{array}{lll} 4 & -21 & s \end{array}\right], \quad A_{2}(\mathbf{b})=\left[\begin{array}{lll} 3 s & 4-6 & 1 \end{array}\right]$$

$$\operatorname{det} A=3 s^{2}-12=3(s+2)(s-2)$$

$$x_{1}=\frac{\operatorname{det} A_{1}(\mathbf{b})}{\operatorname{det} A}=\frac{4 s+2}{3(s+2)(s-2)} \quad x_{2}=\frac{\operatorname{det} A_{2}(\mathbf{b})}{\operatorname{det} A}=\frac{3 s+24}{3(s+2)(s-2)}=\frac{s+8}{(s+2)(s-2)}$$

## 数学代考|线性代数代写LINEAR ALGEBRA代考|A Formula for A-1

Cramer 的规则很容易得出一个反函数的一般公式 $n \times n$ 矩阵 $A$. 这 $j$ 第列 $A^{-1}$ 是一个向量 $\mathbf{x}$ 满足
$$A \mathbf{x}=\mathbf{e} j$$

、left 的分隔符缺失或无法识别

$$\operatorname{det} A i(\mathbf{e} j)=(-1)^{i+j} \operatorname{det} A j i=C_{j i}$$

（4）右侧的辅因子矩阵称为 $A$ ，用 $\operatorname{adj}$ 表示 $A$ （术语伴椭在线性变换的高级文本中也有另一个含义。) 下一个 定理简单地重述 (4)。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

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## 数学代考|线性代数代写LINEAR ALGEBRA代考|Column Operations

We can perform operations on the columns of a matrix in a way that is analogous to the row operations we have considered. The next theorem shows that column operations have the same effects on determinants as row operations.

Remark: The Principle of Mathematical Induction says the following: Let $P(n)$ be a statement that is either true or false for each natural number $n$. Then $P(n)$ is true for all $n \geq 1$ provided that $P(1)$ is true, and for each natural number $k$, if $P(k)$ is true, then $P(k+1)$ is true. The Principle of Mathematical Induction is used to prove the next theorem.
If $A$ is an $n \times n$ matrix, then $\operatorname{det} A^{T}=\operatorname{det} A$.
PROOF The theorem is obvious for $n=1$. Suppose the theorem is true for $k \times k$ determinants and let $n=k+1$. Then the cofactor of $a_{1 j}$ in $A$ equals the cofactor of $a_{j 1}$ in $A^{T}$, because the cofactors involve $k \times k$ determinants. Hence the cofactor expansion of $\operatorname{det} A$ along the first row equals the cofactor expansion of $\operatorname{det} A^{T}$ down the first column. That is, $A$ and $A^{T}$ have equal determinants. The theorem is true for $n=1$, and the truth of the theorem for one value of $n$ implies its truth for the next value of $n$. By the Principle of Mathematical Induction, the theorem is true for all $n \geq 1$.

Because of Theorem 5, each statement in Theorem 3 is true when the word row is replaced everywhere by column. To verify this property, one merely applies the original Theorem 3 to $A^{T}$. A row operation on $A^{T}$ amounts to a column operation on $A$.

Column operations are useful for both theoretical purposes and hand computations. However, for simplicity we’ll perform only row operations in numerical calculations.

## 数学代考|线性代数代写LINEAR ALGEBRA代考|Determinants and Matrix Products

The proof of the following useful theorem is at the end of the section. Applications are in the exercises.
Multiplicative Property
If $A$ and $B$ are $n \times n$ matrices, then $\operatorname{det} A B=(\operatorname{det} A)(\operatorname{det} B)$.
EXAMPLE 5 Verify Theorem 6 for $A=\left[\begin{array}{ll}6 & 1 \ 3 & 2\end{array}\right]$ and $B=\left[\begin{array}{ll}4 & 3 \ 1 & 2\end{array}\right]$ SOLUTION
$$A B=\left[\begin{array}{ll} 6 & 1 \ 3 & 2 \end{array}\right]\left[\begin{array}{ll} 4 & 3 \ 1 & 2 \end{array}\right]=\left[\begin{array}{ll} 25 & 20 \ 14 & 13 \end{array}\right]$$
and
$$\operatorname{det} A B=25 \cdot 13-20 \cdot 14=325-280=45$$
Since $\operatorname{det} A=9$ and $\operatorname{det} B=5$,
$$(\operatorname{det} A)(\operatorname{det} B)=9 \cdot 5=45=\operatorname{det} A B$$
Warning: A common misconception is that Theorem 6 has an analogue for sums of matrices. However, $\operatorname{det}(A+B)$ is not equal to $\operatorname{det} A+\operatorname{det} B$, in general.

## 数学代考|线性代数代写LINEAR ALGEBRA代考|Determinants and Matrix Products

If一个和乙是n×n矩阵，然后这一个乙=(这一个)(这乙).

(这一个)(这乙)=9⋅5=45=这一个乙

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

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## 数学代考|线性代数代写Linear algebra代考|MATH7000 Linear Equations and Electrical Networks

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## 数学代考|线性代数代写LINEAR ALGEBRA代考|Linear Equations and Electrical Networks

Current flow in a simple electrical network can be described by a system of linear equations. A voltage source such as a battery forces a current of electrons to flow through the network. When the current passes through a resistor (such as a lightbulb or motor), some of the voltage is “used up”; by Ohm’s law, this “voltage drop” across a resistor is given by
$$V=R I$$
where the voltage $V$ is measured in volts, the resistance $R$ in ohms (denoted by $\Omega$ ), and the current flow $I$ in amperes (amps, for short).

The network in Figure 1 contains three closed loops. The currents flowing in loops 1,2 , and 3 are denoted by $I_{1}, I_{2}$, and $I_{3}$, respectively. The designated directions of such loop currents are arbitrary. If a current turns out to be negative, then the actual direction of current flow is opposite to that chosen in the figure. If the current direction shown is away from the positive (longer) side of a battery $(-1)$ around to the negative (shorter) side, the voltage is positive; otherwise, the voltage is negative.
Current flow in a loop is governed by the following rule.

## 数学代考|线性代数代写LINEAR ALGEBRA代考|Difference Equations

In many fields such as ecology, economics, and engineering, a need arises to model mathematically a dynamic system that changes over time. Several features of the system are each measured at discrete time intervals, producing a sequence of vectors $\mathbf{x}{0}, \mathbf{x}{1}$, $\mathbf{x}{2}, \ldots$. The entries in $\mathbf{x}{k}$ provide information about the state of the system at the time of the $k$ th measurement.
If there is a matrix $A$ such that $\mathbf{x}{1}=A \mathbf{x}{0}, \mathbf{x}{2}=A \mathbf{x}{1}$, and, in general,
$$\mathbf{x}{k+1}=A \mathbf{x}{k} \quad \text { for } k=0,1,2, \ldots$$
then (5) is called a linear difference equation (or recurrence relation). Given such an equation, one can compute $\mathbf{x}{1}, \mathbf{x}{2}$, and so on, provided $\mathbf{x}{0}$ is known. Sections $4.8$ and $4.9$, and several sections in Chapter 5 , will develop formulas for $\mathbf{x}{k}$ and describe what can happen to $\mathbf{x}_{k}$ as $k$ increases indefinitely. The discussion below illustrates how a difference equation might arise.

A subject of interest to demographers is the movement of populations or groups of people from one region to another. The simple model here considers the changes in the population of a certain city and its surrounding suburbs over a period of years.

Fix an initial year-say, 2014- and denote the populations of the city and suburbs that year by $r_{0}$ and $s_{0}$, respectively. Let $\mathbf{x}_{0}$ be the population vector

For 2015 and subsequent years, denote the populations of the city and suburbs by the vectors
$$\mathbf{x}{1}=\left[\begin{array}{l} r{1} \ s_{1} \end{array}\right], \quad \mathbf{x}{2}=\left[\begin{array}{l} r{2} \ s_{2} \end{array}\right], \quad \mathbf{x}{3}=\left[\begin{array}{l} r{3} \ s_{3} \end{array}\right], \ldots$$
Our goal is to describe mathematically how these vectors might be related.

## 数学代考|线性代数代写LINEAR ALGEBRA代考|Linear Equations and Electrical Networks

$$V=R I$$

## 数学代考|线性代数代写LINEAR ALGEBRA代考|Difference Equations

$$\mathbf{x} k+1=A \mathbf{x} k \quad \text { for } k=0,1,2, \ldots$$

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。