Posted on Categories:Ordinary Differential Equations, 常微分方程, 数学代写

# 数学代写|常微分方程代考Ordinary Differential Equations代写|Exact differential equations

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## 数学代写|常微分方程代考Ordinary Differential Equations代写|Exact differential equations

In the discussion on simple ODEs of type 2 , we have already used the fact that an ODE $y^{\prime}=f(x, y)$ can be also represented as an equation involving differentials as $d y=f(x, y) d x$ or, equivalently, $f(x, y) d x-d y=0$. An exact differential equation extends this concept for specific sets of coefficient functions of $d x$ and $d y$.
The starting point of our discussion is the following equation:
$$g(x, y) d x+h(x, y) d y=0$$
(Also called differential form.) This equation appears as a simple reformulation of the ODE $y^{\prime}=f(x, y)$, in the case where $f(x, y)=-g(x, y) / h(x, y)$. However, this reformulation can be advantageous to extend the validity of the ODE problem.
To illustrate this fact, consider the ODE
$$y^{\prime}+\frac{x}{y}=0$$
The general solutions to this equation are given by $y(x ; c)= \pm \sqrt{c^2-x^2}$, both defined in the open interval $-|c|<x<|c|$. On $x= \pm c$ the equation is not defined since $y=0$ although it is clear that $y=0$ at $x= \pm c$. However, if we now rewrite the differential equation (2.8), we have $x d x+y d y=0$. Now, we recognise that this latter equation corresponds to the differential of $F(x, y)=x^2+y^2=c^2$, which gives, in implicit form, the general solution of our ODE, that is, semicircles centred on the origin and their radius is determined by the constant of integration.

Notice that, for the differential equation (2.8) to be well defined, the functions $g$ and $h$ cannot be both zero at the same $(x, y)$, otherwise the equation becomes indefinite. Therefore, we require
$$g^2+h^2>0$$

## 数学代写|常微分方程代考Ordinary Differential Equations代写|The Cauchy problem and existence of solutions

The theory of ODEs focuses on the issue of existence and regularity of solutions to ordinary differential equations. In this chapter, we illustrate the main ‘classical’ results in this field in the case of scalar equations. These results are very well discussed in the book by Earl Alexander Coddington and Norman Levinson [36] to which we refer for further details.

Now, our main purpose is to prove existence of a real differentiable function $y$ defined on a real interval $I$ such that
$$y^{\prime}=f(x, y(x)), \quad(x, y(x)) \in D, \quad x \in I$$
From a geometrical point of view, a solution $y$ on $I$ is a function whose graph $(x, y(x)), x \in I$, has the slope $f(x, y(x))$ for each $x \in I$.

We assume that $I \subset \mathbb{R}$ denotes an interval that can be closed, open or partially closed, i.e. $[a, b],(a, b),[a, b),(a, b], a<b$. Further, the domain $D$ denotes an open connected set in the $(x, y)$ plane. Following the usual notation, with $C^k(I)$, resp. $C^k(D)$, we denote the set of real-valued functions having continuous derivatives up to order $k$; in $x, \frac{d^k}{d x^k}$, resp. in $x$ and $y, \frac{\partial^k}{\partial x^p \partial y^q}$, $p+q=k$. In the case of closed or partially closed $I$, we refer to one-sided derivatives at the end points. See the Appendix for more details on these function spaces.

# 常微分方程代写

## 数学代写|常微分方程代考Ordinary Differential Equations代写|Exact differential equations

$$g(x, y) d x+h(x, y) d y=0$$
(也称为微分形式。)这个方程是ODE $y^{\prime}=f(x, y)$的一个简单的重新表述，在$f(x, y)=-g(x, y) / h(x, y)$。然而，这种重新表述有利于扩展ODE问题的有效性。

$$y^{\prime}+\frac{x}{y}=0$$

$$g^2+h^2>0$$

## 数学代写|常微分方程代考Ordinary Differential Equations代写|The Cauchy problem and existence of solutions

$$y^{\prime}=f(x, y(x)), \quad(x, y(x)) \in D, \quad x \in I$$

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## MATLAB代写

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