# 微积分网课代修|常微分方程代写Ordinary Differential Equation代考|MATH221 Reparametrization of Time

• 单变量微积分
• 多变量微积分
• 傅里叶级数
• 黎曼积分
• ODE
• 微分学

## 微积分网课代修|常微分方程代写Ordinary Differential Equation代考|Reparametrization of Time

Suppose that $U$ is an open set in $\mathbb{R}^{n}, f: U \rightarrow \mathbb{R}^{n}$ is a smooth function, and $g: U \rightarrow \mathbb{R}$ is a positive smooth function. What is the relationship among the solutions of the differential equations
\begin{aligned} \dot{x} &=f(x), \ \dot{x} &=g(x) f(x) ? \end{aligned}
The vector fields defined by $f$ and $g f$ have the same direction at each point in $U$, only their lengths are different. Thus, by our geometric interpretation of autonomous differential equations, it is intuitively clear that the differential equations (1.10) and (1.11) have the same phase portraits in $U$. This fact is a corollary of the next proposition.

Proposition 1.14. If $J \subset \mathbb{R}$ is an open interval containing the origin and $\gamma: J \rightarrow \mathbb{R}^{n}$ is a solution of the differential equation (1.10) with $\gamma(0)=$ $x_{0} \in U$, then the function $B: J \rightarrow \mathbb{R}$ given by
$$B(t)=\int_{0}^{t} \frac{1}{g(\gamma(s))} d s$$
is invertible on its range $K \subseteq \mathbb{R}$. If $\rho: K \rightarrow J$ is the inverse of $B$, then the identity
$$\rho^{\prime}(t)=g(\gamma(\rho(t))$$
holds for all $t \in K$, and the function $\sigma: K \rightarrow \mathbb{R}^{n}$ given by $\sigma(t)=\gamma(\rho(t))$ is the solution of the differential equation (1.11) with initial condition $\sigma(0)=$ $x_{0}$.

Proof. The function $s \mapsto 1 / g(\gamma(s))$ is continuous on $J$. So $B$ is defined on $J$ and its derivative is everywhere positive. Thus, $B$ is invertible on its range. If $\rho$ is its inverse, then
$$\rho^{\prime}(t)=\frac{1}{B^{\prime}(\rho(t))}=g(\gamma(\rho(t)))$$
and
$$\sigma^{\prime}(t)=\rho^{\prime}(t) \gamma^{\prime}(\rho(t))=g(\gamma(\rho(t)) f(\gamma(\rho(t))=g(\sigma(t)) f(\sigma(t))$$

## 微积分网课代修|常微分方程代写Ordinary Differential Equation代考|Stability and Linearization

Rest points and periodic orbits correspond to very special solutions of autonomous differential equations. However, in the applications these are often the most important orbits. In particular, common engineering practice is to run a process in “steady state.” If the process does not stay near the steady state after a small disturbance, then the control engineer will have to face a difficult problem. We will not solve the control problem here, but we will introduce the mathematical definition of stability and the classic methods that can be used to determine the stability of rest points and periodic orbits.

The concept of Lyapunov stability is meant to capture the intuitive notion of stability-an orbit is stable if solutions that start nearby stay nearby. To give the formal definition, let us consider the autonomous differential equation
$$\dot{x}=f(x)$$
defined on an open set $U \subset \mathbb{R}^{n}$ and its flow $\phi_{t}$.

## 微积分网课代修|常微分方程代写Ordinary Differential Equation代考|Reparametrization of Time

$$\dot{x}=f(x), \dot{x} \quad=g(x) f(x) ?$$

$$B(t)=\int_{0}^{t} \frac{1}{g(\gamma(s))} d s$$

$$\rho^{\prime}(t)=g(\gamma(\rho(t))$$

$$\rho^{\prime}(t)=\frac{1}{B^{\prime}(\rho(t))}=g(\gamma(\rho(t)))$$

$$\sigma^{\prime}(t)=\rho^{\prime}(t) \gamma^{\prime}(\rho(t))=g(\gamma(\rho(t)) f(\gamma(\rho(t))=g(\sigma(t)) f(\sigma(t))$$

## 微积分网课代修|常微分方程代写Ordinary Differential Equation代 考 Stability and Linearization

Lyapunov 稳定性的概念是为了捕捉直观的稳定性概念一一如果从附近开始的解保持在 附近，那么轨道就是稳定的。为了给出正式的定义，让我们考虑自治微分方程
$$\dot{x}=f(x)$$