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^{-...
Massachusetts Institute of Technology professor, Gilbert Strang, explains differential equations and linear algebra which are two crucial subjects in science and engineering. This video series develops those subjects both separately and together and supplements Gil Strang's textbook on this subject....
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...
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)=\...
for then Equation (3.1) becomes a differential equation for u(t) in the interval 1 < t < 2. Since u(1) = g(1), this determines the values of u(t) for 1 ≦ t ≦2, assuming that a, b, f, and g are integrable. In the same way, u(t) can be determined successively in th...
Zerfaoui M, Naura AS, Errami Y, Hans CP, Rezk BM, Park J, Elsegeiny W, Kim H, Lord K, Kim JG, Boulares AH (2009) Effects of PARP-1 deficiency on airway inflammatory cell recruitment in response to LPS or TNF: differential effects on CXCR2 ligands and duffy antigen receptor for ...
具体方法是d(expressions(x,y))dx=d(expressions(x,y))dx=some functions ofxxalone 将微分方程整理成这种方式后,再对两边积分∫∫ 积分因子法 Integration Factor 积分因子法适合解决 First order linear ODE dydx+p(x)y=q(x)dydx+p(x)y=q(x) ...
1 微分方程式 (Differential Equations)微分方程式 微分方程式: 方程式或等式中含有自變數之未知函數及其導函數或微分者稱之 函數(因變數)y = f (x ); x :自變數 ⇒ 一般式或通式()0,=y x F where ()()x f y y x F -=, 函數的微分(differential) dy : 代表函數y 隨著自變數x 之變化...
Continuity Properties : Laws of Continuity 注意第四个:只有当f(x)f(x)连续时,极限符号才能从括号外面移到里面 Continuous Extension to a Point 观察这个函数: 函数sinxxsinxx在x=0x=0上不连续:原因是,虽然limx→0−sinxx=limx→0+sinxx=1limx→0−sinxx=limx→0+sinxx=1,但是sin...
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}...