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# 数学代写|实分析代写Real Analysis代考|MA50400

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## 数学代写|实分析代写Real Analysis代考|Series Solutions in the Second-Order Linear Case

In this section we shall consider, in some detail, series solutions for two kinds of ordinary differential equations.
The first kind is
$$y^{\prime \prime}+P(t) y^{\prime}+Q(t) y=0,$$
where $P(t)$ and $Q(t)$ are given by convergent power-series expansions for $|t|<R$ :
\begin{aligned} & P(t)=a_0+a_1 t+a_2 t^2+\cdots, \ & Q(t)=b_0+b_1 t+b_2 t^2+\cdots . \end{aligned}
We seek power-series solutions of the form
$$y(t)=c_0+c_1 t+c_2 t^2+\cdots .$$
The same methods and theorem that handle this first kind of equation apply also to $n^{\text {th }}$-order homogeneous linear equations and to first-order homogeneous systems when the leading coefficient is 1 and the other coefficients are given by convergent power series. The second-order case, however, is by far the most important for applications and is sufficiently illustrative that we shall limit our attention to it.
The idea in finding the solutions is to assume that we have a convergent powerseries solution $y(t)$ as above, to substitute the series into the equation, and to sort out the conditions that are imposed on the unknown coefficients. Our theorems on power series in Section I.7 guarantee us that the operations of differentiation and multiplication of power series maintain convergence, and thus the result of substituting into the equation is that we obtain an equality of a convergent power series with 0 . Corollary 1.39 then shows that all the coefficients of this last power series must be 0 , and we obtain recursive equations for the unknown coefficients. There is one theorem about the equations under study, and it tells us that the power series for $y(t)$ that we obtain by these manipulations is indeed convergent; we state and prove this theorem shortly.

## 数学代写|实分析代写Real Analysis代考|Measures and Examples

In the theory of the Riemann integral, as discussed in Chapter I for $\mathbb{R}^1$ and in Chapter III for $\mathbb{R}^n$, we saw that Riemann integration is a powerful tool when applied to continuous functions. Riemann integration makes sense also when applied to certain kinds of discontinuous functions, but then the theory has some weaknesses.

Without any change in the definitions, one of these is that the theory applies only to bounded functions. Thus we can compute $\int_0^1 x^p d x=\left[x^{p+1} /(p+1)\right]0^1=$ $(p+1)^{-1}$ for $p \geq 0$, but only the right side makes sense for $-1{n=1}^{\infty} \frac{\cos n \theta}{n}$ and $\frac{1}{2}(\pi-\theta)=\sum_{n=1}^{\infty} \frac{\sin n \theta}{n}$ for $0<\theta<2 \pi$.

When we tried to explain these similar-looking identities with Fourier series, we were able to handle the second one because $\frac{1}{2}(\pi-\theta)$ is a bounded function, but we were not able to handle the first one because $\frac{1}{2} \log \left(\frac{1}{2-2 \cos \theta}\right)$ is unbounded.
Other weaknesses appeared in Chapters I-IV at certain times: when we always had to arrange for the set of integration to be bounded, when we had no clue which sequences $\left{c_n\right}$ of Fourier coefficients occurred in the beautiful formula given by Parseval’s Theorem, when Fubini’s Theorem turned out to be awkward to apply to discontinuous functions, and when the change-of-variables formula did not immediately yield the desired identities even in simple cases like the change from Cartesian coordinates to polar coordinates.

The Lebesgue integral will solve all these difficulties when formed with respect to “Lebesgue measure” in the setting of $\mathbb{R}^n$. In addition, the Lebesgue integral will be meaningful in other settings. For example, the Lebesgue integral will be meaningful on the unit sphere in Euclidean space, while the Riemann integral would always require a choice of coordinates. The Lebesgue integral will be meaningful also in other situations where we can take advantage of some action by a group (such as a rotation group) that is difficult to handle when the setting has to be Euclidean. And the Lebesgue integral will enable us to provide a rigorous foundation for the theory of probability.

## 数学代写|实分析代写Real Analysis代考|Series Solutions in the Second-Order Linear Case

$$y^{\prime \prime}+P(t) y^{\prime}+Q(t) y=0,$$

\begin{aligned} & P(t)=a_0+a_1 t+a_2 t^2+\cdots, \ & Q(t)=b_0+b_1 t+b_2 t^2+\cdots . \end{aligned}

$$y(t)=c_0+c_1 t+c_2 t^2+\cdots .$$

## MATLAB代写

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