Differential Equations Higher-Order Linear Differential Equations Syllabus

Solve $y'' + 4y = 0$ (state the general solution).

Roots $m = \pm 2i$ ($\alpha=0,\beta=2$). General solution: $y = c_{1}\cos 2x + c_{2}\sin 2x$.

What is the purpose of Reduction of Order, and what is required to use it?

To find a second, independent solution $y_{2}$ of a second-order linear ODE given one known nontrivial solution $y_{1}$. It reduces the equation to a first-order one via the substitution $y_{2}=u(x)y_{1}(x)$.

State the reduction-of-order formula for the second solution of $y'' + P(x)y' + Q(x)y = 0$ given $y_{1}$.

$$y_{2} = y_{1}(x)\int \frac{e^{-\int P(x)\,dx}}{y_{1}^{2}(x)}\,dx$$

Use reduction of order: given $y_{1}=e^{x}$ solves $y''-2y'+y=0$, find $y_{2}$.

Here $P=-2$, so $e^{-\int P dx}=e^{2x}$ and $y_{2}=e^{x}\int \frac{e^{2x}}{e^{2x}}dx = e^{x}\int dx = x e^{x}$.

When is the Method of Undetermined Coefficients applicable?

Only for constant-coefficient linear ODEs $L(y)=g(x)$ where $g(x)$ is a constant, polynomial, exponential $e^{\alpha x}$, $\sin\beta x$, $\cos\beta x$, or finite sums/products of these.

In undetermined coefficients, what trial form $y_{p}$ do you assume for $g(x)=$ a polynomial of degree $n$?

$y_{p} = A_{n}x^{n} + A_{n-1}x^{n-1} + \cdots + A_{1}x + A_{0}$ (a full general polynomial of the same degree).

In undetermined coefficients, what trial $y_{p}$ do you assume for $g(x) = e^{\alpha x}$? For $g(x)=\sin\beta x$ or $\cos\beta x$?

For $e^{\alpha x}$: $y_{p}=A e^{\alpha x}$. For $\sin\beta x$ or $\cos\beta x$: $y_{p}=A\cos\beta x + B\sin\beta x$ (always include both sine and cosine).

State the multiplication (modification) rule in undetermined coefficients when the trial $y_{p}$ duplicates a term of $y_{c}$.

Multiply the trial $y_{p}$ by $x^{s}$, where $s$ is the smallest positive integer that removes all duplication with the complementary solution $y_{c}$.

Find the form of $y_{p}$ for $y''-3y'+2y = e^{2x}$ (do not solve for the coefficient).

Since $y_{c}=c_{1}e^{x}+c_{2}e^{2x}$, the term $e^{2x}$ duplicates $y_{c}$, so use $y_{p}=A x e^{2x}$.

When is Variation of Parameters preferred over undetermined coefficients?

When the ODE has variable coefficients, or when $g(x)$ is not of the special form (e.g. $\tan x$, $\sec x$, $\frac{1}{x}$, $\ln x$). It applies to any linear ODE once $y_{c}$ is known.

For $y''+P y'+Q y=f(x)$ with fundamental solutions $y_{1},y_{2}$, give the variation-of-parameters formulas for $u_{1}'$ and $u_{2}'$.

$u_{1}' = -\dfrac{y_{2}f(x)}{W}$ and $u_{2}' = \dfrac{y_{1}f(x)}{W}$, where $W = W(y_{1},y_{2})$. Then $y_{p}=u_{1}y_{1}+u_{2}y_{2}$.

In variation of parameters, what must the equation be put into before reading off $f(x)$?

Standard form with leading coefficient 1: $y'' + P(x)y' + Q(x)y = f(x)$. Divide through by $a_{2}(x)$ first.

Write the full particular solution integral formula from variation of parameters for a 2nd-order ODE.

$$y_{p} = -y_{1}\int \frac{y_{2}f(x)}{W}\,dx + y_{2}\int \frac{y_{1}f(x)}{W}\,dx$$

What is the general form of a (homogeneous) Cauchy-Euler equation of order 2?

$a x^{2}y'' + b x y' + c y = 0$, where $a,b,c$ are constants; each term has $x$ raised to the same power as the order of the derivative.

How do you solve a Cauchy-Euler equation, and what auxiliary equation results from $ax^2y''+bxy'+cy=0$?

Substitute $y = x^{m}$. This gives the auxiliary equation $a m(m-1) + b m + c = 0$, i.e. $a m^{2} + (b-a)m + c = 0$.

For a Cauchy-Euler equation with distinct real roots $m_{1}\neq m_{2}$, what is the general solution?

$y = c_{1}x^{m_{1}} + c_{2}x^{m_{2}}$.

For a Cauchy-Euler equation with a repeated real root $m$, what is the general solution?

$y = c_{1}x^{m} + c_{2}x^{m}\ln x$.

For a Cauchy-Euler equation with complex roots $m = \alpha \pm \beta i$, what is the general solution?

$y = x^{\alpha}\big[c_{1}\cos(\beta \ln x) + c_{2}\sin(\beta \ln x)\big]$.

What substitution transforms a Cauchy-Euler equation into one with constant coefficients?

$x = e^{t}$ (equivalently $t = \ln x$), which converts $ax^{2}y''+bxy'+cy=g(x)$ into a constant-coefficient linear ODE in $t$.

Under $x=e^{t}$, how do the operators $x\frac{dy}{dx}$ and $x^{2}\frac{d^{2}y}{dx^{2}}$ transform (with $t=\ln x$)?

$x\dfrac{dy}{dx} = \dfrac{dy}{dt}$ and $x^{2}\dfrac{d^{2}y}{dx^{2}} = \dfrac{d^{2}y}{dt^{2}} - \dfrac{dy}{dt}$.

State the differential equation governing free undamped (simple harmonic) motion of a mass-spring system.

$m\dfrac{d^{2}x}{dt^{2}} + kx = 0$, or equivalently $x'' + \omega^{2}x = 0$ with $\omega^{2} = \dfrac{k}{m}$.

Give the general solution and angular frequency for free undamped motion $x'' + \omega^{2}x = 0$.

$x(t) = c_{1}\cos\omega t + c_{2}\sin\omega t$, with $\omega = \sqrt{k/m}$. Period $T = \dfrac{2\pi}{\omega}$, frequency $f = \dfrac{\omega}{2\pi}$.

Write free undamped motion in amplitude-phase form and give the amplitude and phase.

$x(t) = A\sin(\omega t + \phi)$ where amplitude $A = \sqrt{c_{1}^{2}+c_{2}^{2}}$ and phase angle $\phi$ satisfies $\tan\phi = \dfrac{c_{1}}{c_{2}}$.

State the differential equation for free damped motion of a mass-spring system.

$m x'' + \beta x' + k x = 0$, often written $x'' + 2\lambda x' + \omega^{2}x = 0$ with $2\lambda = \dfrac{\beta}{m}$ and $\omega^{2} = \dfrac{k}{m}$.

Classify the three cases of damped motion by the discriminant of $x''+2\lambda x'+\omega^{2}x=0$.

Overdamped: $\lambda^{2}-\omega^{2}>0$ (distinct real roots). Critically damped: $\lambda^{2}-\omega^{2}=0$ (repeated root). Underdamped: $\lambda^{2}-\omega^{2}<0$ (complex roots, oscillatory decay).

Give the solution form for the underdamped case of $x''+2\lambda x'+\omega^{2}x=0$.

$x(t) = e^{-\lambda t}\big(c_{1}\cos\sqrt{\omega^{2}-\lambda^{2}}\,t + c_{2}\sin\sqrt{\omega^{2}-\lambda^{2}}\,t\big)$ — decaying oscillation with quasi-frequency $\sqrt{\omega^{2}-\lambda^{2}}$.

Give the solution form for the critically damped case of $x''+2\lambda x'+\omega^{2}x=0$.

$x(t) = e^{-\lambda t}(c_{1} + c_{2}t)$ — no oscillation; the fastest non-oscillatory return to equilibrium.

State the differential equation for driven (forced) motion with damping and an external force $f(t)$.

$m x'' + \beta x' + k x = f(t)$. The response is $x = x_{c}$ (transient) $+\, x_{p}$ (steady-state).

Distinguish the transient and steady-state terms in forced damped motion.

The transient $x_{c}$ contains the decaying exponential $e^{-\lambda t}$ and dies out as $t\to\infty$; the steady-state $x_{p}$ persists and matches the frequency of the driving force $f(t)$.

Explain pure resonance in undamped forced motion $x''+\omega^{2}x = F_{0}\cos\gamma t$.

Resonance occurs when the driving frequency equals the natural frequency, $\gamma = \omega$. Then $y_{p}$ requires the factor $t$, giving $x_{p}\propto t\sin\omega t$, whose amplitude grows without bound as $t\to\infty$.

State the differential equation for the charge $q(t)$ on the capacitor in an LRC series circuit.

$L\dfrac{d^{2}q}{dt^{2}} + R\dfrac{dq}{dt} + \dfrac{1}{C}q = E(t)$, where $L$ = inductance, $R$ = resistance, $C$ = capacitance, $E(t)$ = impressed voltage.

How does the LRC-circuit equation relate to the mass-spring damped equation (analogy of terms)?

$L \leftrightarrow m$ (mass), $R \leftrightarrow \beta$ (damping), $\dfrac{1}{C} \leftrightarrow k$ (spring constant), $E(t) \leftrightarrow f(t)$ (driving force), and charge $q \leftrightarrow$ displacement $x$.

Give the conditions for overdamped, critically damped, and underdamped behavior of the free LRC circuit ($E(t)=0$) in terms of $R,L,C$.

Overdamped: $R^{2} - \dfrac{4L}{C} > 0$. Critically damped: $R^{2} - \dfrac{4L}{C} = 0$. Underdamped: $R^{2} - \dfrac{4L}{C} < 0$.

How is current $i(t)$ obtained from charge $q(t)$ in an LRC circuit?

$i(t) = \dfrac{dq}{dt}$. The current satisfies the same form of ODE obtained by differentiating: $L i'' + R i' + \dfrac{1}{C}i = E'(t)$.

Compare Undetermined Coefficients and Variation of Parameters: coefficient requirement and allowed $g(x)$.

Undetermined Coefficients: constant coefficients only, $g(x)$ restricted to polynomials/exponentials/sines/cosines. Variation of Parameters: any coefficients (variable OK) and any continuous $g(x)$, but requires integrals and known $y_{c}$.