Differential Equations Higher-Order Linear Differential Equations Flashcards

24 sample cards from the Higher-Order Linear Differential Equations deck

Sampled from the end of the deck, so these are different cards from the ones shown on the syllabus page.

1. 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.

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

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

3. 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.

4. 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. $a m^{2} + (b-a)m + c = 0$.

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

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

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

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

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

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

8. 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 $t$.

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

10. 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 $\omega^{2} = \dfrac{k}{m}$.

11. 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 $f = \dfrac{\omega}{2\pi}$.

12. 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 $\tan\phi = \dfrac{c_{1}}{c_{2}}$.

13. 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 $\omega^{2} = \dfrac{k}{m}$.

14. 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).

15. 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 $\sqrt{\omega^{2}-\lambda^{2}}$.

16. 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.

17. 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).

18. 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 $f(t)$.

19. 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 $t\to\infty$.

20. 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.

21. 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 $x$.

22. 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: $R^{2} - \dfrac{4L}{C} < 0$.

23. 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: $L i'' + R i' + \dfrac{1}{C}i = E'(t)$.

24. 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 $y_{c}$.

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Higher-Order Linear Differential Equations flashcards FAQ