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# 数学代写|线性代数代写Linear algebra代考|MAST10007 APPLICATIONS TO DIFFERENTIAL EQUATIONS

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## 数学代写|线性代数代写Linear algebra代考|APPLICATIONS TO DIFFERENTIAL EQUATIONS

This section describes continuous analogues of the difference equations studied in Section 5.6. In many applied problems, several quantities are varying continuously in time, and they are related by a system of differential equations:
\begin{aligned} x_1^{\prime} &=a_{11} x_1+\cdots+a_{1 n} x_n \ x_2^{\prime} &=a_{21} x_1+\cdots+a_{2 n} x_n \ & \vdots \ x_n^{\prime} &=a_{n 1} x_1+\cdots+a_{n n} x_n \end{aligned}
Here $x_1, \ldots, x_n$ are differentiable functions of $t$, with derivatives $x_1^{\prime}, \ldots, x_n^{\prime}$, and the $a_{i j}$ are constants. The crucial feature of this system is that it is linear. To see this, write the system as a matrix differential equation
$$\mathbf{x}^{\prime}(t)=A \mathbf{x}(t)$$
where
$$\mathbf{x}(t)=\left[\begin{array}{c} x_1(t) \ \vdots \ x_n(t) \end{array}\right], \quad \mathbf{x}^{\prime}(t)=\left[\begin{array}{c} x_1^{\prime}(t) \ \vdots \ x_n^{\prime}(t) \end{array}\right], \quad \text { and } \quad A=\left[\begin{array}{ccc} a_{11} & \cdots & a_{1 n} \ \vdots & & \vdots \ a_{n 1} & \cdots & a_{n n} \end{array}\right]$$
A solution of equation (1) is a vector-valued function that satisfies (1) for all $t$ in some interval of real numbers, such as $t \geq 0$.

Equation (1) is linear because both differentiation of functions and multiplication of vectors by a matrix are linear transformations. Thus, if $\mathbf{u}$ and $\mathbf{v}$ are solutions of $\mathbf{x}^{\prime}=A \mathbf{x}$, then $c \mathbf{u}+d \mathbf{v}$ is also a solution, because
\begin{aligned} (c \mathbf{u}+d \mathbf{v})^{\prime} &=c \mathbf{u}^{\prime}+d \mathbf{v}^{\prime} \ &=c A \mathbf{u}+d A \mathbf{v}=A(c \mathbf{u}+d \mathbf{v}) \end{aligned}

(Engineers call this property superposition of solutions.) Also, the identically zero function is a (trivial) solution of (1). In the terminology of Chapter 4, the set of all solutions of (1) is a subspace of the set of all continuous functions with values in $\mathbb{R}^n$.

## 数学代写|线性代数代写Linear algebra代考|Decoupling a Dynamical System

The following discussion shows that the method of Examples 1 and 2 produces a fundamental set of solutions for any dynamical system described by $\mathbf{x}^{\prime}=A \mathbf{x}$ when $A$ is $n \times n$ and has $n$ linearly independent eigenvectors, that is, when $A$ is diagonalizable. Suppose the eigenfunctions for $A$ are
$$\mathbf{v}_1 e^{\lambda_1 t}, \quad \ldots, \mathbf{v}_n e^{\lambda_n t}$$
with $\mathbf{v}_1, \ldots, \mathbf{v}_n$ linearly independent eigenvectors. Let $P=\left[\begin{array}{lll}\mathbf{v}_1 & \cdots & \mathbf{v}_n\end{array}\right]$, and let $D$ be the diagonal matrix with entries $\lambda_1, \ldots, \lambda_n$, so that $A=P D P^{-1}$. Now make a change of variable, defining a new function $\mathbf{y}$ by
$$\mathbf{y}(t)=P^{-1} \mathbf{x}(t) \quad \text { or, equivalently, } \quad \mathbf{x}(t)=P \mathbf{y}(t)$$
The equation $\mathbf{x}(t)=P \mathbf{y}(t)$ says that $\mathbf{y}(t)$ is the coordinate vector of $\mathbf{x}(t)$ relative to the eigenvector basis. Substitution of $P \mathbf{y}$ for $\mathbf{x}$ in the equation $\mathbf{x}^{\prime}=A \mathbf{x}$ gives
$$\frac{d}{d t}(P \mathbf{y})=A(P \mathbf{y})=\left(P D P^{-1}\right) P \mathbf{y}=P D \mathbf{y}$$
Since $P$ is a constant matrix, the left side of (5) is $P \mathbf{y}^{\prime}$. Left-multiply both sides of (5) by $P^{-1}$ and obtain $\mathbf{y}^{\prime}=D \mathbf{y}$, or
$$\left[\begin{array}{c} y_1^{\prime}(t) \ y_2^{\prime}(t) \ \vdots \ y_n^{\prime}(t) \end{array}\right]=\left[\begin{array}{cccc} \lambda_1 & 0 & \cdots & 0 \ 0 & \lambda_2 & & \vdots \ \vdots & & \ddots & 0 \ 0 & \cdots & 0 & \lambda_n \end{array}\right]\left[\begin{array}{c} y_1(t) \ y_2(t) \ \vdots \ y_n(t) \end{array}\right]$$

## 数学代写|线性代数代写Linear algebra代考|APPLICATIONS TO DIFFERENTIAL EQUATIONS

$$x_1^{\prime}=a_{11} x_1+\cdots+a_{1 n} x_n x_2^{\prime} \quad=a_{21} x_1+\cdots+a_{2 n} x_n \vdots x_n^{\prime} \quad=a_{n 1} x_1+\cdots+a_{n n} x_n$$

$$\mathbf{x}^{\prime}(t)=A \mathbf{x}(t)$$

$$(c \mathbf{u}+d \mathbf{v})^{\prime}=c \mathbf{u}^{\prime}+d \mathbf{v}^{\prime} \quad=c A \mathbf{u}+d A \mathbf{v}=A(c \mathbf{u}+d \mathbf{v})$$

## 数学代写线性代数代写Linear algebra代考|Decoupling a Dynamical System

$$\mathbf{v}_1 e^{\lambda_1 t}, \quad \ldots, \mathbf{v}_n e^{\lambda_n t}$$

$$\mathbf{y}(t)=P^{-1} \mathbf{x}(t) \quad \text { or, equivalently, } \quad \mathbf{x}(t)=P \mathbf{y}(t)$$

$$\frac{d}{d t}(P \mathbf{y})=A(P \mathbf{y})=\left(P D P^{-1}\right) P \mathbf{y}=P D \mathbf{y}$$

## MATLAB代写

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