7 Differential Equations and Dynamic Models Free Online FlashCards

What does a differential equation describe?

A differential equation relates an unknown function to one or more derivatives, describing how a quantity changes rather than specifying the quantity directly.

What defines an initial-value problem?

An initial-value problem consists of a differential equation together with an initial condition, such as $y(0)=y_0$.

What information does a slope field show?

A slope field places a short segment with slope $f(t,y)$ at each point $(t,y)$, showing the local directions solution curves follow.

How are equilibria found in $y'=g(y)$?

For $y'=g(y)$, an equilibrium solution is a constant value satisfying $g(y)=0$. A system starting there remains there.

Where does $y'=y(4-y)$ increase?

For $y'=y(4-y)$, solutions increase when $0<y<4$ and decrease when $y<0$ or $y>4$.

When is a differential equation separable?

A separable equation can be written as $\frac{dy}{dt}=f(t)g(y)$, allowing the variables to be isolated: $\frac{1}{g(y)}dy=f(t)dt$.

Why check constant solutions before separating?

Find constant solutions before dividing by $g(y)$, because dividing by it can remove solutions where $g(y)=0$.

Solve $y'=2ty$ with $y(0)=3$.

For $\frac{dy}{dt}=2ty$ with $y(0)=3$, the solution is $y(t)=3e^{t^2}$.

What is the solution of $P'=rP$?

The exponential model $\frac{dP}{dt}=rP$ has solution $P(t)=P_0e^{rt}$, where $r$ is the constant proportional growth rate.

What are the doubling-time and half-life formulas?

For exponential growth, the doubling time is $\frac{\ln 2}{r}$ when $r>0$. For decay with constant $k>0$, the half-life is $\frac{\ln 2}{k}$.

What does the carrying capacity $K$ represent?

The logistic model is $\frac{dP}{dt}=rP\left(1-\frac{P}{K}\right)$, where $K$ is the carrying capacity limiting long-term growth.

At what population is logistic growth fastest?

For the logistic model with $0<P<K$, growth is fastest at $P=\frac{K}{2}$, the inflection point of the S-shaped solution.