# 数学代写|常微分方程代写ordinary differential equation代考|MATH221

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## 数学代写|常微分方程代写ordinary differential equation代考|Important “Named” Definite Integrals with Variable Limits

You should be familiar with a number of “named” functions (such as the natural logarithm and the arctangent) that can be given by definite integrals. For the two examples just cited,
$$\ln (x)=\int_1^x \frac{1}{s} d s \quad \text { for } x>0$$
and
$$\arctan (x)=\int_0^x \frac{1}{1+s^2} d s .$$
While $\ln (x)$ and $\arctan (x)$ can be defined independently of these integrals, their alternative definitions do not provide us with particularly useful ways to compute these functions by hand (unless $x$ is something special, such as 1 ). Indeed, if you need the value of $\ln (x)$ or $\arctan (x)$ for, say, $x=18$, then you are most likely to “compute” these values by having your calculator or computer or published tables ${ }^2$ tell you the (approximate) value of $\ln (18)$ or $\arctan (18)$. Thus, for computational purposes, we might as well just view $\ln (x)$ and $\arctan (x)$ as names for the above integrals, and be glad that their values can easily be looked up electronically or in published tables.

It turns out that other integrals arise often enough in applications that workers dealing with these applications have decided to “name” these integrals, and to have their values tabulated. Two noteworthy “named integrals” are:

• The error function, denoted by erf and given by
$$\operatorname{erf}(x)=\int_0^x \frac{2}{\sqrt{\pi}} e^{-s^2} d s .$$
• The sine-integral function, denoted by $\mathrm{Si}$ and given by ${ }^3$
$$\operatorname{Si}(x)=\int_0^x \frac{\sin (s)}{s} d s .$$

## 数学代写|常微分方程代写ordinary differential equation代考|Constant (or Equilibrium) Solutions

There is one type of particular solution that is easily determined for many first-order differential equations using elementary algebra: the “constant” solution.

A constant solution to a given differential equation is simply a constant function that satisfies that differential equation. Remember, $y$ is a constant function if its value, $y(x)$, is some fixed constant for all $x$; that is, for some single number $y_0$,
$$y(x)=y_0 \quad \text { for all } x .$$
Such solutions are also sometimes called equilibrium solutions. In an application involving some process that can vary with $x$, these solutions describe situations in which the process does not vary with $x$. This often means that all the factors influencing the process are “balancing out”, leaving the process in a “state of equilibrium”. As we will later see, this sometimes means that these solutions – whether called constant or equilibrium — are the most important solutions to a given differential equation. 1
IDExample 3.1: Consider the differential equation
$$\frac{d y}{d x}=2 x y^2-4 x y$$
and the constant function
$$y(x)=2 \quad \text { for all } x$$

Since the derivative of a constant function is zero, plugging in this function, $y=2$ into
$$\frac{d y}{d x}=2 x y^2-4 x y$$
gives
$$0=2 x \cdot 2^2-4 x \cdot 2,$$
which, after a little arithmetic and algebra, reduces further to
$$0=0$$

# 常微分方程代写

## 数学代写|常微分方程代写ordinary differential equation代考|Important “Named” Definite Integrals with Variable Limits

$$\ln (x)=\int_1^x \frac{1}{s} d s \quad \text { for } x>0$$

$$\arctan (x)=\int_0^x \frac{1}{1+s^2} d s$$

• 误差函数，由 erf 表示并由下式给出
$$\operatorname{erf}(x)=\int_0^x \frac{2}{\sqrt{\pi}} e^{-s^2} d s .$$
• 正弦积分函数，表示为 $\mathrm{Si}$ 并由 ${ }^3$
$$\operatorname{Si}(x)=\int_0^x \frac{\sin (s)}{s} d s .$$

## 数学代写|常微分方程代写ordinary differential equation代考|Constant (or Equilibrium) Solutions

$$y(x)=y_0 \quad \text { for all } x .$$

IDExample 3.1：考虑微分方程
$$\frac{d y}{d x}=2 x y^2-4 x y$$

$$y(x)=2 \quad \text { for all } x$$

$$\frac{d y}{d x}=2 x y^2-4 x y$$

$$0=2 x \cdot 2^2-4 x \cdot 2$$

$$0=0$$

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