TongZhuo's Blog

Video Link 【Segment 1: Industry Dominance】 For years, Nvidia and Intel have dominated the semiconductor industry. Nvidia’s graphics processors have been the backbone of AI training, running everything from chatbots to self-driving cars. While Intel’s chips have powered devices from personal computers to massive data centers. But now, a new contender is emerging, and it’s shaking up the entire landscape: Cambricon Technologies, a Beijing-based chip designer that’s rapidly gaining...

为什么特解的差是齐次方程的解？ 1. 理论基础 对于 线性非齐次微分方程： y′+a(x)y=b(x)（非齐次方程）y' + a(x)y = b(x) \quad \text{（非齐次方程）} y′+a(x)y=b(x)（非齐次方程） 若已知两个特解 y1y_1y1​ 和 y2y_2y2​，则它们的差 y0=y2−y1y_0 = y_2 - y_1y0​=y2​−y1​ 必为对应的 齐次方程： y′+a(x)y=0（齐次方程）y' + a(x)y = 0 \quad \text{（齐次方程）} y′+a(x)y=0（齐次方程） 的解。 2. 数学推导 • 代入 y1y_1y1​ 和 y2y_2y2​ 到非齐次方程： {y1′+a(x)y1=b(x)y2′+a(x)y2=b(x)\begin{cases} y_1' + a(x)y_1 = b(x) \\ y_2' + a(x)y_2 = b(x) \end{cases} {y1′​+a(x)y1​=b(x)y2′​+a(x)y2​=b(x)​ •...

单词释义 petition - n./v. paraphrase：formal request signed by many people synonym：appeal, plea example：They started a petition to save the park, which gathered over 1,000 signatures within a week. appropriate - adj./v. paraphrase：① (adj.) suitable; ② (v.) take something without permission synonym：① fitting, proper; ② seize, take example：The teacher chose an appropriate book for the class, but some students found it too easy. harvest - n./v. paraphrase：gather crops synonym：collect,...

type one：几何关系列出等式解方程 第一道： 第二道： 对于第二题，思路大概是这个思路，但是有一点别扭的就是字母的表示，可以把P点的坐标设为(a,b),其他字母还沿用以前的习惯 这种类型不难，关键在于吃透题目想要表达的等式是啥 type two：整体代换解方程 关键在于令u=yxu=\frac{y}{x}u=xy​,只要想到这一步这种类型就好解了 type three：根据线性微分方程解的结构构造方程组解未知数 第一个式子p−q=0p - q = 0p−q=0 令： py1−qy2=k(y1−y2)(其中 k 为常数),py_1 - qy_2 = k(y_1 - y_2) \quad (\text{其中 } k \text{ 为常数}), py1​−qy2​=k(y1​−y2​)(其中 k 为常数), 则需满足系数关系： p=k,−q=−k⇒p=q.p = k, \quad -q = -k \quad \Rightarrow \quad p = q. p=k,−q=−k⇒p=q. 第二个式子p+q=1p + q = 1p+q=1 题目还要求...

释义列表 index - n. paraphrase：a list of topics or names in a book, often at the end synonym：list, guide example：She checked the index of the textbook to find the chapter about climate change, but it wasn’t listed there. occupation - n. paraphrase：a job or activity you do regularly synonym：job, profession example：His occupation as a teacher kept him busy, though he sometimes missed working outdoors. implication - n. paraphrase：a possible result or meaning of...

希望明天不会忘，希望明天不会忘，希望明天不会忘 例题 解 第一步：写出特征方程求解 二阶导是平方，一阶导是一次方，一个单独的y就是常数项 r2−r=0r^2-r=0 r2−r=0 解出来r1=0,r2=1r_1=0,r_2=1r1​=0,r2​=1 然后根据这个表写出通解 Y=C1+C2exY = C_1 + C_2 e^x Y=C1​+C2​ex 特征方程 r2+pr+q=0r^2 + pr + q = 0r2+pr+q=0 的根 微分方程 y′′+py′+qy=0y'' + py' + qy = 0y′′+py′+qy=0 的通解 ​两个不等实根 r1≠r2r_1 \neq r_2r1​=r2​ y=C1er1x+C2er2xy = C_1 e^{r_1 x} + C_2 e^{r_2 x}y=C1​er1​x+C2​er2​x ​两个相等实根 r1=r2r_1 = r_2r1​=r2​ y=(C1+C2x)er1xy = (C_1 + C_2 x) e^{r_1 x}y=(C1​+C2​x)er1​x ​共轭复根 r1,2=a±iβr_{1,2}...

释义列表 significant - adj. paraphrase ：very important or noticeable synonym ：important, major example ：The discovery was significant because it solved a problem scientists had studied for years. craft - n. paraphrase ：skill in making things by hand synonym ：art, skill example ：She used her craft to create pottery, which she sold at the local market every weekend. delay - v. paraphrase ：make something happen later synonym ：postpone, stall example ：The flight was delayed due to bad...

公式太长了，根本记不住，现在只用记一个公式μ(x)=e∫P(x)dx\mu(x) = e^{\int P(x)dx}μ(x)=e∫P(x)dx(积分因子，AI起的名字) 1. 写出标准形式 原方程： dzdx+6xz=x\frac{dz}{dx} + \frac{6}{x}z = x dxdz​+x6​z=x 符合一阶线性微分方程的标准形式： z′+P(x)z=Q(x)z' + P(x)z = Q(x) z′+P(x)z=Q(x) 其中： • P(x)=6xP(x) = \frac{6}{x}P(x)=x6​ • Q(x)=xQ(x) = xQ(x)=x 2. 计算积分因子 积分因子公式： μ(x)=e∫P(x)dx=e∫6xdx=e6ln⁡∣x∣=x6\mu(x) = e^{\int P(x)dx} = e^{\int \frac{6}{x}dx} = e^{6\ln|x|} = x^6 μ(x)=e∫P(x)dx=e∫x6​dx=e6ln∣x∣=x6 3. 方程两边乘以积分因子 用 μ(x)=x6\mu(x) = x^6μ(x)=x6...

type one：对称性的利用 第一个例子： 拆分积分区间 原积分为对称区间 ∫−π4π411−sin⁡xdx\int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \frac{1}{1 - \sin x} dx∫−4π​4π​​1−sinx1​dx，将其拆分为两部分： ∫−π4011−sin⁡xdx+∫0π411−sin⁡xdx.\int_{-\frac{\pi}{4}}^{0} \frac{1}{1 - \sin x} dx + \int_{0}^{\frac{\pi}{4}} \frac{1}{1 - \sin x} dx. ∫−4π​0​1−sinx1​dx+∫04π​​1−sinx1​dx. 对负区间进行变量替换 对左侧积分 ∫−π4011−sin⁡xdx\int_{-\frac{\pi}{4}}^{0} \frac{1}{1 - \sin x} dx∫−4π​0​1−sinx1​dx，令 x=−tx = -tx=−t（即 t=−xt = -xt=−x），则： • 积分上下限变为：x=−π4→t=π4x =...

1. sin⁡nx\sin^n xsinnx和cos⁡nx\cos^n xcosnx在[0,π2][0, \frac{\pi}{2}][0,2π​]上的积分 • nnn为偶数时： ∫0π2sin⁡nxdx=∫0π2cos⁡nxdx=(n−1)!!n!!⋅π2\int_{0}^{\frac{\pi}{2}} \sin^n x dx = \int_{0}^{\frac{\pi}{2}} \cos^n x dx = \frac{(n-1)!!}{n!!} \cdot \frac{\pi}{2} ∫02π​​sinnxdx=∫02π​​cosnxdx=n!!(n−1)!!​⋅2π​ 例如： ∫0π2sin⁡4xdx=34⋅12⋅π2=3π16\int_{0}^{\frac{\pi}{2}} \sin^4 x dx = \frac{3}{4} \cdot \frac{1}{2} \cdot \frac{\pi}{2} = \frac{3\pi}{16} ∫02π​​sin4xdx=43​⋅21​⋅2π​=163π​ •...