Differential Equations First-Order Ordinary Differential Equations Flashcards

24 sample cards from the First-Order Ordinary Differential Equations deck

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

1. What does a direction field (slope field) of $y'=f(x,y)$ represent?

   At each point $(x,y)$ it draws a short segment with slope $f(x,y)$; solution curves are tangent to these segments everywhere, giving a qualitative picture of solutions without solving.

2. What is an isocline in a direction field?

   An isocline is the curve $f(x,y)=k$ along which all direction-field segments have the same slope $k$; plotting isoclines helps sketch the field efficiently.

3. Define an autonomous first-order differential equation.

   One in which the independent variable does not appear explicitly: $\frac{dy}{dx}=f(y)$. Its slope depends only on $y$.

4. What are critical points (equilibria) of the autonomous equation $y'=f(y)$?

   The constant solutions $y=c$ where $f(c)=0$. They correspond to the zeros of $f$ and appear as horizontal solution lines.

5. On a phase line, how do you classify a critical point as asymptotically stable, unstable, or semi-stable?

   Stable (attractor): arrows point toward it on both sides. Unstable (repeller): arrows point away on both sides. Semi-stable: arrows point toward it on one side and away on the other.

6. Using the derivative test, when is a critical point $c$ of $y'=f(y)$ asymptotically stable?

   If $f(c)=0$ and $f'(c)<0$, then $c$ is asymptotically stable; if $f'(c)>0$ it is unstable. ($f'(c)=0$ is inconclusive.)

7. Write the Picard iteration formula for the IVP $y'=f(x,y),\ y(x_{0})=y_{0}$.

   Equivalent integral equation $y(x)=y_{0}+\int_{x_{0}}^{x} f(t,y(t))\,dt$; iterate $y_{n+1}(x)=y_{0}+\int_{x_{0}}^{x} f\big(t,y_{n}(t)\big)\,dt$ starting from $y_{0}(x)=y_{0}$.

8. What does the method of successive approximations (Picard iteration) establish theoretically?

   Under a Lipschitz condition on $f$ in $y$, the iterates $y_{n}$ converge uniformly to the unique solution of the IVP, providing a constructive proof of existence and uniqueness.

9. Compute the first two Picard iterates for $y'=y,\ y(0)=1$.

   $y_{0}=1$; $y_{1}=1+\int_{0}^{x}1\,dt=1+x$; $y_{2}=1+\int_{0}^{x}(1+t)\,dt=1+x+\frac{x^{2}}{2}$, approaching $e^{x}$.

10. State the growth/decay model and its solution.

    $\frac{dP}{dt}=kP,\ P(0)=P_{0}$ gives $P(t)=P_{0}e^{kt}$. $k>0$ is exponential growth, $k<0$ is decay.

11. For radioactive decay with half-life $T$, relate the decay constant $k$ to $T$.

    From $P(t)=P_{0}e^{kt}$ with $k<0$, $\tfrac12 P_{0}=P_{0}e^{kT}$ gives $k=-\frac{\ln 2}{T}$, i.e. $T=\frac{\ln 2}{|k|}$.

12. State Newton's law of cooling as a differential equation.

    $\frac{dT}{dt}=k\,(T-T_{m})$, where $T$ is the object's temperature, $T_{m}$ the constant ambient temperature, and $k<0$ a constant.

13. Give the solution of Newton's law of cooling $\frac{dT}{dt}=k(T-T_{m}),\ T(0)=T_{0}$.

    $T(t)=T_{m}+(T_{0}-T_{m})e^{kt}$; as $t\to\infty$, $T\to T_{m}$.

14. Set up the differential equation for a mixing (compartment) problem for the amount of salt $A(t)$.

    $\frac{dA}{dt}=R_{in}-R_{out}=(\text{inflow rate}\times\text{inflow conc.})-(\text{outflow rate}\times\frac{A}{V(t)})$, where $V(t)$ is the current volume in the tank.

15. In a mixing problem, how is the tank volume $V(t)$ determined when inflow and outflow rates differ?

    $V(t)=V_{0}+(r_{in}-r_{out})\,t$, where $r_{in},r_{out}$ are the volumetric flow rates; $V$ is constant only when $r_{in}=r_{out}$.

16. Write the logistic population model differential equation.

    $\frac{dP}{dt}=rP\left(1-\frac{P}{K}\right)$, where $r$ is the intrinsic growth rate and $K$ the carrying capacity.

17. Give the solution of the logistic equation with $P(0)=P_{0}$.

    $P(t)=\dfrac{K P_{0}}{P_{0}+(K-P_{0})e^{-rt}}$, so $P(t)\to K$ as $t\to\infty$.

18. What are the equilibrium solutions of the logistic equation and their stability?

    $P=0$ (unstable) and $P=K$ (asymptotically stable). Populations with $0<P_{0}<K$ increase toward $K$; those above $K$ decrease toward $K$.

19. At what population size does the logistic model predict the fastest growth (maximum $\frac{dP}{dt}$)?

    At the inflection point $P=\frac{K}{2}$, where $\frac{d^{2}P}{dt^{2}}=0$ and the growth rate is greatest.

20. Compare separable, linear, and exact first-order equations by their identifying form.

    Separable: $y'=g(x)h(y)$ (variables split). Linear: $y'+P(x)y=Q(x)$ (first power of $y$, use $\mu=e^{\int P}$). Exact: $M\,dx+N\,dy=0$ with $M_{y}=N_{x}$ (potential $F$ exists).

21. How can you recognize and solve $\frac{dy}{dx}=f(ax+by+c)$?

    Substitute $u=ax+by+c$; then $\frac{du}{dx}=a+b\,f(u)$, which is separable in $u$ and $x$.

22. What is the interval of validity (interval of definition) of a solution to an IVP?

    The largest open interval containing $x_{0}$ on which the particular solution is defined, continuous, and satisfies the ODE; it may be smaller than the domain of $f$.

23. State the superposition-free general solution structure of a linear first-order equation in terms of complementary and particular parts.

    $y=y_{c}+y_{p}$, where $y_{c}=ce^{-\int P\,dx}$ solves the homogeneous equation $y'+Py=0$ and $y_{p}$ is any particular solution of $y'+Py=Q$.

24. For the autonomous logistic-type harvesting model $\frac{dP}{dt}=rP(1-\frac{P}{K})-h$, what determines whether the population can survive?

    Whether $f(P)=rP(1-\frac{P}{K})-h$ has real positive critical points; if the constant harvest $h$ exceeds the maximum growth $\frac{rK}{4}$, no equilibrium exists and the population declines to extinction.

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First-Order Ordinary Differential Equations flashcards FAQ