# 微积分网课代修|常微分方程代写Ordinary Differential Equation代考|MATH350 Existence and Uniqueness

• 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|Existence and Uniqueness

Let $J \subseteq \mathbb{R}, U \subseteq \mathbb{R}^{n}$, and $\Lambda \subseteq \mathbb{R}^{k}$ be open subsets, and suppose that $f: J \times U \times \Lambda \rightarrow \mathbb{R}^{n}$ is a smooth function. Here the term “smooth” means that the function $f$ is continuously differentiable. An ordinary differential equation (ODE) is an equation of the form
$$\dot{x}=f(t, x, \lambda)$$
where the dot denotes differentiation with respect to the independent variable $t$ (usually a measure of time), the dependent variable $x$ is a vector of state variables, and $\lambda$ is a vector of parameters. As convenient terminology,especially when we are concerned with the components of a vector differential equation, we will say that equation (1.1) is a system of differential equations. Also, if we are interested in changes with respect to parameters, then the differential equation is called a family of differential equations.

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

Differential equations may be classified in several different ways. In this section we note that the independent variable may be implicit or explicit, and that higher order derivatives may appear.
An autonomous differential equation is given by
$$\dot{x}=f(x, \lambda), \quad x \in \mathbb{R}^{n}, \quad \lambda \in \mathbb{R}^{k} ;$$
that is, the function $f$ does not depend explicitly on the independent variable. If the function $f$ does depend explicitly on $t$, then the corresponding differential equation is called nonautonomous.

In physical applications, we often encounter equations containing second, third, or higher order derivatives with respect to the independent variable. These are called second order differential equations, third order differential equations, and so on, where the the order of the equation refers to the order of the highest order derivative with respect to the independent variable that appears explicitly. In this hierarchy, a differential equation is called a first order differential equation.

Recall that Newton’s second law-the rate of change of the linear momentum acting on a body is equal to the sum of the forces acting on the body-involves the second derivative of the position of the body with respect to time. Thus, in many physical applications the most common differential equations used as mathematical models are second order differential equations. For example, the natural physical derivation of van der Pol’s equation leads to a second order differential equation of the form
$$\ddot{u}+b\left(u^{2}-1\right) \dot{u}+\omega^{2} u=a \cos \Omega t .$$

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

$$\dot{x}=f(t, x, \lambda)$$

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

$$\dot{x}=f(x, \lambda), \quad x \in \mathbb{R}^{n}, \quad \lambda \in \mathbb{R}^{k} ;$$

$$\ddot{u}+b\left(u^{2}-1\right) \dot{u}+\omega^{2} u=a \cos \Omega t .$$
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