求解y' = −2xy 且y(0) = 2 解: y\neq 0 \frac{dy}{dx}=-2xy\Rightarrow\frac{1}{y}dy=-2xdx ,两边同时积分得到 \int\frac{1}{y}dy=\int(-2x)dx+C\Rightarrow \ln|y| = −x^2 +C 最终有 |y|=e^C e^{-x^2} \Rightarrow y= \pm e^C e^{-x^2}=C_1 e^{-...
A Fourier series separates a periodic function F(x) into a combination (infinite) of all basis functions cos(nx) and sin(nx). 13:55 Video length is 13:55 8.1b: Examples of Fourier Series Even functions use only cosines (F(–x) = F(x)) and odd functions use only sines. The co...
例: (eax)′=a⋅eax ; (lnx2)′=1x2⋅2x 高阶导数的运算 (x^{a})^{(n)}=x^{a-n} \prod_{k=0}^{n-1} (a-k) (\frac{1}{x})^{(n)}=(-1)^n\frac{n!}{x^{n+1}} \quad\quad (n=1,2,...)(\ln x)^{(n)}=(-1)^{(n-1)}\frac{(n-1)!}{x^{n}...
4.1.1 A first change of variables It is possible to make the change of variable xi = log(Si), i = 1, …, d; calling x the vector (x1,…, xd) and P˜ the function, P˜(x,t) = P(S, t), the final value problem (4.5) (4.6) becomes (4.8)(∂P˜∂t+L˜P...
The line x =12h is such a line of symmetry for our tube flow. In order to exploit this symmetry, we introduce the new analytic function (2)w(ζ)=g13i[z(ζ+i)−12h], which is real for all real values of the complex potential ζ. In terms of w, the free boundary condition (...
of the form (4+1)/(x-1)(Bx+c)/(x^2+3) with values for A, B, C , with no extra fractions Integrates 1/(x^2+3) to obtain ktan^(-1)α/(√3)Obtains a completely correct expression for the general solution:xy=ln(x-1)+√3tan^(-1)x/(√3)+C Condone omission of c ....
The unknown in a differential equation is a function, and a differential equation will almost always involve this function and one or more derivatives of the function. For example, f 0.x/ D f .x/ is a simple differential equation (asking if there is any function f such that it equals ...
()()x x y x x y x y dx dy y x x ∆-∆+=∆∆=='→∆→∆00lim lim ※ 一個函數的自變量的增量∆x 趨近某一極限時，其因變量的增量∆y 與自變量的增量∆x 之商的極限 ※ 函數y 對自變數x 之瞬間變化率(instantaneous rate of change); 函數y 的無窮小的變化量dy (微分)...
2 Chapter 1/ First-Order Differential Equations and Models 1.1 A Modeling Adventure Comments In this introductory section we hit upon several of the main themes of our approach to modeling and differential equations. We construct an initial value problem that models a changing fish population. The...
线性无关,若方程 c1y1(x)+c2y2(x)+···+cnyn(x)=0 仅在c1=c2=···=cn=0 时成立 若不是线性无关那么就是线性相关 Wronskian of n functions n个函数的Wronskian 若p_1, p_2, ··· , p_n ∈ C(I) ,那么n个函数的Wronskian为 W\left(f_1, f_2, \cdots, f_n\right)(x)=\...