Second-Order Linear ODEs: Homogeneous二阶线性常微分方程：齐次方程
University-Style Practice Problems大学风格练习题
MEDIUMHARDCOREPROOFAPPLIED
Sections 1 to 6: superposition, characteristic equation (distinct real, complex, repeated roots), Wronskian and fundamental sets, reduction of orderCALC IV第 1 至 6 节：叠加原理、特征方程（不同实根、复数根、重复根）、朗斯基行列式与基本解组、降阶法CALC IV
Name:姓名：Date:日期：
PART I · CORE TECHNIQUESComputational fluency · 28 marks计算能力 · 28分
The Characteristic Equation特征方程
For each ODE, write the characteristic equation, identify which root case applies, and state the general solution. Show the discriminant $b^{2}-4ac$ explicitly. Verify your solution by substituting back into the ODE before applying initial conditions.对每个常微分方程，写出特征方程，判断适用的根的情形，并写出通解。明确展示判别式 $b^{2}-4ac$。在代入初始条件前，将解代回方程进行验证。
Q1MEDIUMCOREcharacteristic equation: distinct real roots特征方程：不同实根[8 marks]
Solve each initial-value problem (IVP). State the characteristic equation and both roots in each case.求解下列各初值问题（IVP）。写出每种情形的特征方程及两个根。
In each part, identify $\lambda$ and $\mu$ from $r = \lambda \pm i\mu$, write the real general solution using $e^{\lambda t}\cos\mu t$ and $e^{\lambda t}\sin\mu t$, then apply the given initial conditions.在每部分中，从 $r = \lambda \pm i\mu$ 中确定 $\lambda$ 和 $\mu$，用 $e^{\lambda t}\cos\mu t$ 和 $e^{\lambda t}\sin\mu t$ 写出实数形式的通解，然后代入给定的初始条件。
Q3MEDIUMCOREWronskian and fundamental sets朗斯基行列式与基本解组[6 marks]
Compute the Wronskian $W(y_1,y_2)=y_1 y_2'-y_2 y_1'$ for each pair and state whether the pair forms a fundamental set for the given equation.计算每对函数的朗斯基行列式 $W(y_1,y_2)=y_1 y_2'-y_2 y_1'$，并判断该对函数是否构成给定方程的基本解组。
(a)Find the general solution by solving the characteristic equation. Confirm that the second solution $te^{rt}$ is not a scalar multiple of $e^{rt}$ by computing the Wronskian of the pair.通过求解特征方程得到通解。通过计算这对函数的朗斯基行列式，确认第二个解 $te^{rt}$ 不是 $e^{rt}$ 的数量倍。[3]
(b)Apply the initial conditions to determine $c_1$ and $c_2$, and write the particular solution. Verify by substitution that your answer satisfies the IVP.代入初始条件确定 $c_1$ 和 $c_2$，写出特解。通过代入验证你的答案满足该初值问题。[3]
PART II · DEFINITIONS AND PROOFRigorous arguments · 26 marks严格论证 · 26分
Superposition, the Wronskian, and Reduction of Order叠加原理、朗斯基行列式与降阶法
These items are graded on the logic of the argument. State every hypothesis you invoke. In derivations, show each algebraic step; a correct answer with missing steps earns at most half credit.这些题目按论证逻辑评分。陈述你所引用的每个假设。在推导过程中，展示每个代数步骤；答案正确但步骤缺失，最多只能得一半分数。
Q5MEDIUMPROOFprinciple of superposition叠加原理[6 marks]
Let $L[y] = y'' + p(t)y' + q(t)y$ where $p,q$ are continuous on an interval $I$.设 $L[y] = y'' + p(t)y' + q(t)y$，其中 $p,q$ 在区间 $I$ 上连续。
(a)Prove that $L$ is a linear operator: that is, show $L[c_1 y_1 + c_2 y_2] = c_1 L[y_1] + c_2 L[y_2]$ for any differentiable $y_1$, $y_2$ and constants $c_1$, $c_2$.证明 $L$ 是线性算子，即对任意可微函数 $y_1$、$y_2$ 及常数 $c_1$、$c_2$，证明 $L[c_1 y_1 + c_2 y_2] = c_1 L[y_1] + c_2 L[y_2]$。[3]
(b)Hence deduce: if $y_1$ and $y_2$ are both solutions of $L[y]=0$, then $y = c_1 y_1 + c_2 y_2$ is also a solution for any constants $c_1$, $c_2$. State precisely which property of $L$ you used and why it would fail for a nonlinear ODE such as $y'' + (y')^{2} = 0$.由此推断：若 $y_1$ 和 $y_2$ 都是 $L[y]=0$ 的解，则对任意常数 $c_1$、$c_2$，$y = c_1 y_1 + c_2 y_2$ 也是解。准确陈述所用的 $L$ 的哪个性质，并说明为何对非线性方程（如 $y'' + (y')^{2} = 0$）该性质不成立。[3]
Q6HARDPROOFWronskian, linear independence, and Abel's identity朗斯基行列式、线性无关性与阿贝尔恒等式[10 marks]
Let $y_1, y_2$ be solutions of $y'' + p(t)y' + q(t)y = 0$ on an interval $I$ where $p,q$ are continuous.设 $y_1, y_2$ 是 $y'' + p(t)y' + q(t)y = 0$ 在区间 $I$ 上的解，其中 $p,q$ 连续。
(a)Show that if $y_1, y_2$ are linearly dependent on $I$, then $W(y_1,y_2)(t) = 0$ for all $t \in I$. (Hint: write $y_2 = k y_1$ and compute directly.)证明：若 $y_1, y_2$ 在 $I$ 上线性相关，则对所有 $t \in I$ 都有 $W(y_1,y_2)(t) = 0$。（提示：令 $y_2 = k y_1$ 并直接计算。）[2]
(b)Derive Abel's identity: show that $W' = -p(t)W$ and hence $W(t) = W(t_0)\,e^{-\int_{t_0}^{t}p(s)\,ds}$. Conclude that $W$ is either identically zero or never zero on $I$.推导阿贝尔恒等式：证明 $W' = -p(t)W$，从而得出 $W(t) = W(t_0)\,e^{-\int_{t_0}^{t}p(s)\,ds}$。结论是 $W$ 在 $I$ 上恒为零或处处不为零。[5]
(c)Use Abel's identity to find, up to a multiplicative constant, the Wronskian of any two solutions of $y'' + 4y' + 4y = 0$, without solving the equation. Then verify by computing $W$ from the actual solutions.利用阿贝尔恒等式，在不求解方程的情况下，确定 $y'' + 4y' + 4y = 0$ 的任意两个解的朗斯基行列式（精确到一个乘法常数）。然后通过实际解计算 $W$ 加以验证。[3]
Q7HARDPROOFreduction of order: second solution from a known one降阶法：从已知解求第二个解[10 marks]
The equation $t^{2}y'' - 3ty' + 4y = 0$ ($t > 0$) has a known solution $y_1 = t^{2}$.方程 $t^{2}y'' - 3ty' + 4y = 0$（$t > 0$）有一个已知解 $y_1 = t^{2}$。
(a)Verify by substitution that $y_1 = t^{2}$ is indeed a solution.通过代入验证 $y_1 = t^{2}$ 确实是该方程的解。[2]
(b)Set $y_2 = v(t)\,t^{2}$, substitute into the equation divided by $t^{2}$, and show that the equation for $v$ reduces to a first-order linear ODE in $w = v'$. State the ODE clearly.令 $y_2 = v(t)\,t^{2}$，代入除以 $t^{2}$ 后的方程，证明关于 $v$ 的方程可化简为关于 $w = v'$ 的一阶线性常微分方程。清晰写出该方程。[4]
(c)Solve for $w$ and integrate to find $v$; hence write down $y_2$ and confirm that $\{y_1, y_2\}$ is a fundamental set by computing $W(y_1,y_2)$.求解 $w$ 并积分得到 $v$；写出 $y_2$，并通过计算 $W(y_1,y_2)$ 确认 $\{y_1, y_2\}$ 是基本解组。[4]
PART III · APPLICATIONS AND SYNTHESISExtended problems · 28 marks综合应用题 · 28分
Initial-Value Problems, Amplitude-Phase Form, and Free Vibration初值问题、振幅相位形式与自由振动
Set up each problem cleanly. Express solutions in exact form. For amplitude-phase conversions, use $R\cos(\mu t - \phi)$ with $R>0$ and $\phi \in [0, 2\pi)$. In physical problems, interpret the solution in words as well as formulas.清晰建立每道题的模型。以精确形式表示解。振幅相位转换时，使用 $R\cos(\mu t - \phi)$，其中 $R>0$ 且 $\phi \in [0, 2\pi)$。在物理问题中，除公式外还需用文字解释解的含义。
Q8HARDAPPLIEDcomplex-root IVP rewritten in amplitude-phase form复数根初值问题改写为振幅相位形式[10 marks]
(a)Find the characteristic roots and write the general solution $y = e^{\lambda t}(c_1 \cos\mu t + c_2 \sin\mu t)$.求特征根，写出通解 $y = e^{\lambda t}(c_1 \cos\mu t + c_2 \sin\mu t)$。[2]
(b)Apply the initial conditions to find $c_1$ and $c_2$.代入初始条件求 $c_1$ 和 $c_2$。[3]
(c)Rewrite the particular solution in amplitude-phase form $R e^{\lambda t}\cos(\mu t - \phi)$. Compute $R$ and $\phi$ exactly, and state the quasi-period.将特解改写为振幅相位形式 $R e^{\lambda t}\cos(\mu t - \phi)$。精确计算 $R$ 和 $\phi$，并写出准周期。[3]
(d)State whether the system is underdamped, critically damped, or overdamped, and briefly explain what happens to $y(t)$ as $t \to \infty$.判断系统是欠阻尼、临界阻尼还是过阻尼，并简要说明当 $t \to \infty$ 时 $y(t)$ 的行为。[2]
Q9HARDAPPLIEDfree-vibration model: overdamped and critically damped cases自由振动模型：过阻尼与临界阻尼情形[10 marks]
A damped mass-spring system satisfies $my'' + by' + ky = 0$. In each part, determine the root case, solve the IVP, and describe the qualitative long-term behaviour of the displacement $y(t)$.有阻尼弹簧质量系统满足 $my'' + by' + ky = 0$。在每部分中，判断根的情形，求解初值问题，并描述位移 $y(t)$ 的定性长期行为。
(a) $m=1$, $b=5$, $k=4$: $\;y(0)=2$, $\;y'(0)=0$ (overdamped). Find the general solution, apply the ICs, and show that $y(t) \to 0$ monotonically as $t \to \infty$.（过阻尼）。求通解，代入初始条件，并证明当 $t \to \infty$ 时 $y(t)$ 单调趋向 $0$。[5]
(b) $m=1$, $b=4$, $k=4$: $\;y(0)=1$, $\;y'(0)=-5$ (critically damped). Find the particular solution and show that it crosses zero exactly once for $t>0$. Identify the crossing time.（临界阻尼）。求特解，证明其在 $t>0$ 时恰好过零一次。求过零时刻。[5]
Q10HARDAPPLIEDboundary-value problem and the role of the Wronskian边值问题与朗斯基行列式的作用[8 marks]
Consider the boundary-value problem (BVP) $y'' + 4y = 0$ on $[0, L]$.考虑 $[0, L]$ 上的边值问题（BVP）：$y'' + 4y = 0$。
(a)Find the general solution.求通解。[2]
(b)Apply the boundary conditions $y(0)=0$ and $y(L)=0$. Show that the BVP has a nontrivial solution if and only if $L = n\pi/2$ for some positive integer $n$, and find the corresponding nontrivial solutions.代入边界条件 $y(0)=0$ 和 $y(L)=0$。证明当且仅当 $L = n\pi/2$（$n$ 为正整数）时，边值问题有非平凡解，并求出相应的非平凡解。[4]
(c)Contrast with an IVP: explain in one or two sentences why the Wronskian argument that guarantees a unique solution to an IVP does not apply to a BVP.与初值问题对比：用一两句话说明，保证初值问题有唯一解的朗斯基行列式论证为何不适用于边值问题。[2]