Sections 1 to 7: reduction to systems, matrix form, real distinct eigenvalues, complex eigenvalues, repeated eigenvalues, phase portraits and stability, matrix exponential and variation of parameters第 1 至 7 节：化为方程组、矩阵形式、实不同特征值、复特征值、重复特征值、相平面与稳定性、矩阵指数与常数变易法CALC IV
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PART I · CORE TECHNIQUES第一部分 · 核心技巧Computational fluency · 28 marks计算熟练度 · 28 分
Reduction, Eigenvectors, and Real Solutions化简、特征向量与实数解
Show all working. For eigenvalue problems, write the characteristic equation explicitly before factoring. For IVPs, verify your answer by substituting back.展示完整解题过程。特征值问题须明确写出特征方程再分解因式。初值问题须将答案代回验证。
Q1MEDIUMCOREreducing a higher-order ODE to a first-order system将高阶常微分方程化为一阶方程组[8 marks]
Reduce $y''' - 6y'' + 11y' - 6y = e^{t}$ to a first-order system; find the eigenvalues of the companion matrix; state why they equal the characteristic roots.将 $y''' - 6y'' + 11y' - 6y = e^{t}$ 化为一阶方程组，求伴随矩阵的特征值，并说明为何它们等于特征根。
(a)Set $x_1=y$, $x_2=y'$, $x_3=y''$ and write $\mathbf{x}'=A\mathbf{x}+\mathbf{g}(t)$ in full matrix form. State $A$ and $\mathbf{g}(t)$ explicitly.令 $x_1=y$，$x_2=y'$，$x_3=y''$，将 $\mathbf{x}'=A\mathbf{x}+\mathbf{g}(t)$ 写成完整矩阵形式，明确给出 $A$ 和 $\mathbf{g}(t)$。[3]
(b)Find the characteristic polynomial of $A$ and compute its roots (the eigenvalues of $A$).求 $A$ 的特征多项式，并计算其根（即 $A$ 的特征值）。[3]
(c)Explain in one or two sentences why the eigenvalues of the companion matrix equal the characteristic roots of the scalar ODE.用一两句话解释伴随矩阵的特征值为何等于标量常微分方程的特征根。[2]
Q2MEDIUMCOREeigenvalue method: real distinct eigenvalues and IVP特征值法：实不同特征值与初值问题[8 marks]
For $A = \begin{pmatrix}1&3\\3&1\end{pmatrix}$, $\mathbf{x}(0)=(4,2)^T$: find eigenvalues and eigenvectors; solve the IVP; classify the portrait.对于 $A = \begin{pmatrix}1&3\\3&1\end{pmatrix}$，$\mathbf{x}(0)=(4,2)^T$：求特征值和特征向量，求解初值问题，并对相平面进行分类。
(a)Find the eigenvalues and a corresponding eigenvector for each.求各特征值及其对应的一个特征向量。[4]
(b)Write the general solution and apply the initial condition to find the particular solution.写出通解，代入初始条件求特解。[2]
(c)Classify the phase portrait and describe the long-time behaviour of $\mathbf{x}(t)$.对相平面进行分类，并描述 $\mathbf{x}(t)$ 的长时行为。[2]
Q3HARDCOREcomplex eigenvalues: real solution pair and spiral classification复特征值：实数解对与螺旋型分类[6 marks]
Solve $\mathbf{x}' = \begin{pmatrix}-1&2\\-2&-1\end{pmatrix}\mathbf{x}$; write two real solutions; classify the portrait.求解 $\mathbf{x}' = \begin{pmatrix}-1&2\\-2&-1\end{pmatrix}\mathbf{x}$，写出两个实数解，并对相平面进行分类。
(a)Find the complex eigenpair $\lambda=\alpha\pm i\beta$ and the real parts $\mathbf{a},\mathbf{b}$ of the eigenvector. Write the two real linearly independent solutions $\mathbf{x}_1(t)$ and $\mathbf{x}_2(t)$ explicitly.求复特征值对 $\lambda=\alpha\pm i\beta$ 及特征向量的实部 $\mathbf{a},\mathbf{b}$，明确写出两个实线性无关解 $\mathbf{x}_1(t)$ 和 $\mathbf{x}_2(t)$。[4]
(b)Using only $(p,q,\Delta)$, classify the portrait type and state whether the origin is stable.仅用 $(p,q,\Delta)$ 对相平面类型进行分类，并说明原点是否稳定。[2]
Q4HARDCORErepeated defective eigenvalue and generalized eigenvector重亏损特征值与广义特征向量[6 marks]
Solve $\mathbf{x}' = \begin{pmatrix}-2&1\\0&-2\end{pmatrix}\mathbf{x}$; show the matrix is defective; find the generalized eigenvector and general solution.求解 $\mathbf{x}' = \begin{pmatrix}-2&1\\0&-2\end{pmatrix}\mathbf{x}$，证明矩阵有亏损，求广义特征向量和通解。
(a)Find the eigenvalue(s) and show explicitly that the matrix is defective (state geometric vs algebraic multiplicity).求特征值，并明确说明矩阵有亏损（陈述几何重数与代数重数之间的关系）。[2]
(b)Find the generalized eigenvector $\mathbf{w}$ satisfying $(A-\lambda I)\mathbf{w}=\mathbf{v}$, write the general solution, and classify the portrait.求满足 $(A-\lambda I)\mathbf{w}=\mathbf{v}$ 的广义特征向量 $\mathbf{w}$，写出通解，并对相平面进行分类。[4]
PART II · DEFINITIONS AND PROOF第二部分 · 定义与证明Rigorous arguments · 26 marks严密论证 · 26 分
Derivations and Theoretical Results推导与理论结果
These items are graded on the logic of the argument. Quote every theorem you apply. Proofs must state hypotheses, proceed step by step, and justify each cancellation or limit.本部分按论证逻辑评分。须引用所用的每个定理。证明须陈述假设条件，逐步推进，并说明每个消去或极限的理由。
Q5HARDPROOFderiving the eigenvalue method: why $\mathbf{x}=e^{\lambda t}\mathbf{v}$ reduces to $A\mathbf{v}=\lambda\mathbf{v}$推导特征值法：为何 $\mathbf{x}=e^{\lambda t}\mathbf{v}$ 化为 $A\mathbf{v}=\lambda\mathbf{v}$[8 marks]
(a)Substitute the ansatz $\mathbf{x}(t)=e^{\lambda t}\mathbf{v}$ (where $\mathbf{v}\ne\mathbf{0}$ is a constant vector) into $\mathbf{x}'=A\mathbf{x}$ and derive the eigenvalue equation $A\mathbf{v}=\lambda\mathbf{v}$. State clearly why the factor $e^{\lambda t}$ may be cancelled.将试探解 $\mathbf{x}(t)=e^{\lambda t}\mathbf{v}$（$\mathbf{v}\ne\mathbf{0}$ 为常向量）代入 $\mathbf{x}'=A\mathbf{x}$，推导特征方程 $A\mathbf{v}=\lambda\mathbf{v}$，并明确说明为何可以消去因子 $e^{\lambda t}$。[4]
(b)Prove that eigenvectors corresponding to distinct eigenvalues are linearly independent. (Induction on the number of eigenvectors; apply $(A-\lambda_k I)$ to a supposed dependence relation.)证明对应不同特征值的特征向量线性无关。（对特征向量个数进行归纳，将 $(A-\lambda_k I)$ 作用于假定的相关性关系。）[4]
Q6HARDPROOFAbel's theorem for systems and the Wronskian方程组的 Abel 定理与 Wronskian 行列式[8 marks]
Let $\Psi(t)$ be a fundamental matrix for $\mathbf{x}'=A\mathbf{x}$ ($A$ constant $n\times n$) and $W(t)=\det\Psi(t)$.设 $\Psi(t)$ 为 $\mathbf{x}'=A\mathbf{x}$（$A$ 为常数 $n\times n$ 矩阵）的基本矩阵，$W(t)=\det\Psi(t)$。
(a)Show that $W'=(\operatorname{tr}A)\,W$, and derive Abel's formula $W(t)=W(t_0)\exp(\operatorname{tr}(A)(t-t_0))$. (Use the multi-linearity of the determinant, the fact that each column satisfies $\mathbf{x}'=A\mathbf{x}$, and explain why only diagonal terms survive.)证明 $W'=(\operatorname{tr}A)\,W$，并推导 Abel 公式 $W(t)=W(t_0)\exp(\operatorname{tr}(A)(t-t_0))$。（利用行列式的多线性、每列满足 $\mathbf{x}'=A\mathbf{x}$ 的事实，并解释为何只有对角项存留。）[5]
(b)Use Abel's formula to prove: $W(t)\ne 0$ for all $t$ if and only if $W(t_0)\ne 0$ for some (equivalently, every) $t_0$.利用 Abel 公式证明：对所有 $t$ 均有 $W(t)\ne 0$ 当且仅当对某（等价地，每一个）$t_0$ 有 $W(t_0)\ne 0$。[3]
Q7HARDPROOFphase-portrait classification and the stability criterion相平面分类与稳定性判据[10 marks]
Let $p=\operatorname{tr}A$, $q=\det A$, and $\Delta=p^2-4q$ for a $2\times 2$ real constant matrix $A$.设 $p=\operatorname{tr}A$，$q=\det A$，$\Delta=p^2-4q$，$A$ 为 $2\times 2$ 实常数矩阵。
(a)Classify the phase portrait at the origin for each matrix using only $(p,q,\Delta)$. State the portrait type (saddle, stable/unstable spiral, center, stable/unstable node, etc.) and whether the origin is stable.仅用 $(p,q,\Delta)$ 对每个矩阵在原点处的相平面进行分类，说明相平面类型（鞍点、稳定/不稳定螺旋、中心、稳定/不稳定结点等）以及原点是否稳定。[6]
(i) $A=\begin{pmatrix}2&-1\\5&-4\end{pmatrix}$
(ii) $A=\begin{pmatrix}-3&2\\-2&-1\end{pmatrix}$
(iii) $A=\begin{pmatrix}1&-2\\2&1\end{pmatrix}$
(iv) $A=\begin{pmatrix}0&-2\\2&0\end{pmatrix}$
(b)Prove that the origin of $\mathbf{x}'=A\mathbf{x}$ is asymptotically stable if and only if $p<0$ and $q>0$. (Consider the real and complex eigenvalue cases separately; use $\lambda_1+\lambda_2=p$ and $\lambda_1\lambda_2=q$.)证明 $\mathbf{x}'=A\mathbf{x}$ 的原点渐近稳定当且仅当 $p<0$ 且 $q>0$。（分实特征值和复特征值两种情况讨论；利用 $\lambda_1+\lambda_2=p$ 及 $\lambda_1\lambda_2=q$。）[4]
PART III · APPLICATIONS AND SYNTHESIS第三部分 · 应用与综合Extended problems · 28 marks综合题 · 28 分
Applied Systems and the Matrix Exponential应用方程组与矩阵指数
For mixing problems, define variables and set up the differential equations from physical principles before writing the matrix form. For variation of parameters, state the formula before integrating.混合问题须先根据物理原理定义变量、建立微分方程，再写成矩阵形式。常数变易法须先写出公式再积分。
Q8HARDAPPLIEDfull IVP by the eigenvalue method: complex eigenvalues, spiral, trajectory description完整初值问题：复特征值、螺旋型、轨迹描述[10 marks]
Solve the IVP $\mathbf{x}'=\begin{pmatrix}-1&2\\-2&-1\end{pmatrix}\mathbf{x}$, $\mathbf{x}(0)=(1,0)^T$. Use the same matrix as Q3.求解初值问题 $\mathbf{x}'=\begin{pmatrix}-1&2\\-2&-1\end{pmatrix}\mathbf{x}$，$\mathbf{x}(0)=(1,0)^T$。矩阵与第 3 题相同。
(a)Find the eigenpair (you may cite your work from Q3 if completed; otherwise redo it here). Write the two real linearly independent solutions.求特征值对（如已完成第 3 题可引用该结果；否则在此重新推导）。写出两个实线性无关解。[4]
(b)Apply the initial condition to find $c_1$ and $c_2$.代入初始条件，求 $c_1$ 和 $c_2$。[2]
(c)Differentiate $\mathbf{x}(t)$ and verify component-by-component that it satisfies $\mathbf{x}'=A\mathbf{x}$.对 $\mathbf{x}(t)$ 求导，逐分量验证其满足 $\mathbf{x}'=A\mathbf{x}$。[2]
(d)Classify the phase portrait and describe the trajectory starting at $(1,0)^T$ in words: spiral direction, speed, limiting behaviour.对相平面进行分类，用文字描述从 $(1,0)^T$ 出发的轨迹：螺旋方向、速度及极限行为。[2]
Q9HARDAPPLIEDtwo-compartment mixing: system setup, eigenvalue solution, long-time behaviour双容器混合：方程组建立、特征值求解、长时行为[10 marks]
Two interconnected tanks. Tank 1 loses salt at rate $3x_1$ and gains from Tank 2 at rate $x_2$; Tank 2 gains from Tank 1 at rate $2x_1$ and loses at rate $2x_2$. Initially $x_1(0)=3$ kg, $x_2(0)=0$.两个相连的储液槽。储液槽 1 以速率 $3x_1$ 流失盐分，并以速率 $x_2$ 从储液槽 2 补充；储液槽 2 以速率 $2x_1$ 从储液槽 1 补充，并以速率 $2x_2$ 流失。初始时 $x_1(0)=3$ kg，$x_2(0)=0$。
(a)Write the system in matrix form $\mathbf{x}'=A\mathbf{x}$. State $A$ explicitly. Use $(p,q,\Delta)$ to pre-classify the portrait type before computing eigenvalues.将方程组写成矩阵形式 $\mathbf{x}'=A\mathbf{x}$，明确给出 $A$。在计算特征值之前，利用 $(p,q,\Delta)$ 预判相平面类型。[3]
(b)Find the eigenvalues and a corresponding eigenvector for each. Verify each eigenvector by direct multiplication.求各特征值及其对应的一个特征向量，并通过直接相乘验证每个特征向量。[4]
(c)Apply the initial condition and find the particular solution.代入初始条件，求特解。[2]
(d)Describe the long-time behaviour of $x_1(t)$ and $x_2(t)$ as $t\to\infty$. Which mode dominates, and what does it say about the ratio $x_2/x_1$?描述 $t\to\infty$ 时 $x_1(t)$ 和 $x_2(t)$ 的长时行为。哪个模态占主导？这对 $x_2/x_1$ 的比值说明了什么？[1]
Q10HARDAPPLIEDmatrix exponential by diagonalization and variation of parameters for a forced system对角化求矩阵指数与强迫方程组的常数变易法[8 marks]
Let $A=\begin{pmatrix}0&1\\-2&-3\end{pmatrix}$.设 $A=\begin{pmatrix}0&1\\-2&-3\end{pmatrix}$。
(a)Diagonalize $A$: find $P$ and $D$ such that $A=PDP^{-1}$. Compute $P^{-1}$ and verify $PP^{-1}=I$.对 $A$ 进行对角化：求 $P$ 和 $D$ 使得 $A=PDP^{-1}$，计算 $P^{-1}$ 并验证 $PP^{-1}=I$。[3]
(b)Use the diagonalization to compute $e^{At}=Pe^{Dt}P^{-1}$ and verify $e^{A\cdot 0}=I$.利用对角化计算 $e^{At}=Pe^{Dt}P^{-1}$，并验证 $e^{A\cdot 0}=I$。[2]
(c)Using variation of parameters, solve the forced IVP $\mathbf{x}'=A\mathbf{x}+\begin{pmatrix}2\\0\end{pmatrix}$, $\mathbf{x}(0)=\mathbf{0}$, via $\mathbf{x}(t)=\int_0^t e^{A(t-s)}\mathbf{g}\,ds$. Verify your answer at $t=0$.用常数变易法，通过 $\mathbf{x}(t)=\int_0^t e^{A(t-s)}\mathbf{g}\,ds$ 求解强迫初值问题 $\mathbf{x}'=A\mathbf{x}+\begin{pmatrix}2\\0\end{pmatrix}$，$\mathbf{x}(0)=\mathbf{0}$，并在 $t=0$ 处验证答案。[3]