Unit 7: Differential Equations

Differential equation

An equation that relates an unknown function to one or more of its derivatives (describes how a quantity changes).

Solution (to a differential equation)

A function that makes the differential equation true when substituted into the equation along with its derivatives.

Family of solutions

A set of solutions usually represented with one or more arbitrary constants (many functions satisfy the same differential equation).

General solution

A solution containing an arbitrary constant (or constants), representing a whole family of solution curves.

Particular solution

One specific solution from a family, found by using an additional condition such as an initial condition.

Initial condition

A given value of the function at a specific input (e.g., y(a)=b) used to determine the constant(s).

Initial value problem (IVP)

A differential equation paired with an initial condition, used to select a unique particular solution.

Verification (checking a solution)

The process of differentiating a proposed function, substituting into the differential equation, and confirming both sides match.

Derivative notation dy/dx

Notation meaning “the derivative of y with respect to x,” i.e., the instantaneous rate of change of y as x changes.

Prime notation (y′)

An alternative notation for the first derivative dy/dx.

Second derivative (d²y/dx² or y″)

The derivative of the derivative; describes concavity/acceleration of the solution curve.

Form dy/dx = f(x,y)

A common way to write a first-order differential equation, emphasizing slope depends on both x and y.

Domain restriction (in solutions)

A limitation on where a solution is valid (e.g., intervals that do not cross x=0 if dividing by x occurs).

Slope field (direction field)

A diagram showing small line segments at points (x,y) with slope f(x,y) for dy/dx=f(x,y).

Direction segment

A short line drawn at (x,y) in a slope field with slope equal to the differential equation’s value f(x,y).

Tangent requirement (slope fields)

A solution curve passing through a point must be tangent to the slope field segment drawn at that point.

Autonomous differential equation

A differential equation where $\frac{dy}{dx}$ depends only on $y$ (e.g., $\frac{dy}{dx}=g(y)$), not explicitly on $x$.

Horizontal row pattern (autonomous slope fields)

In dy/dx=g(y), all points with the same y-value have the same slope, so slopes repeat across horizontal lines.

Isocline

A curve in the xy-plane along which $\frac{dy}{dx}$ has a constant value (same slope everywhere on that curve).

Equilibrium solution

A constant solution $y=c$ where the derivative is zero ($\frac{dy}{dx}=0$), so the solution stays flat.

Stability (equilibrium)

Describes whether nearby solutions move toward an equilibrium (stable) or away from it (unstable).

Stable equilibrium

An equilibrium where nearby solutions approach it over time (seen in slope fields as “flowing toward” the equilibrium).

Unstable equilibrium

An equilibrium where nearby solutions move away from it over time (seen as “flowing away” in the slope field).

Qualitative analysis

Studying solution behavior (increasing/decreasing, leveling off, long-term trends) without finding an explicit formula.

Euler’s Method

A numerical method that approximates a solution to an IVP using repeated tangent-line (linear) steps.

Step size (h)

The x-increment used in Euler’s Method; smaller h usually improves accuracy but requires more steps.

Euler update formula

The recursion $x_{n+1}=x_n+h$ and $y_{n+1}=y_n+h \,f(x_n,y_n)$ for $\frac{dy}{dx}=f(x,y)$.

Euler table

A table of repeated Euler computations, typically listing xn, yn, f(xn,yn), and the next y-value.

Piecewise linear approximation

The type of curve Euler’s Method creates by connecting successive points with straight-line segments.

Euler method underestimation/overestimation (concavity link)

If the true solution is concave up, Euler’s tangent steps often underestimate; if concave down, they often overestimate (use as a diagnostic).

Separable differential equation

A differential equation that can be rewritten with all y-terms on one side and all x-terms on the other, enabling integration.

Separation of variables

Algebraic rearrangement of dy/dx=f(x)g(y) into (1/g(y))dy = f(x)dx before integrating.

SIPPY method

A mnemonic for separable DEs: Separate, Integrate, Plus C, Plug in initial condition, Y equals (solve for y).

Constant of integration (C)

The arbitrary constant added after integrating; later determined using an initial condition for an IVP.

Implicit solution

A solution left in a form involving both x and y (not solved explicitly for y), often acceptable on exams.

Definite-integral separation approach

Using bounds after separating variables to relate y(a)=ya to y(b)=yb without solving for C first.

Exponential growth/decay differential equation

The model $\frac{dP}{dt} = kP$, where the rate of change is proportional to the amount present.

Continuous growth rate (k)

The constant in $\frac{dP}{dt}=kP$; $k>0$ gives growth, $k<0$ gives decay, with units “per unit time.”

Exponential growth/decay solution

The solution to $\frac{dP}{dt}=kP$: $P(t)=Ce^{kt}$, or $P(t)=P_0e^{kt}$ if $P(0)=P_0$.

Doubling time

For exponential growth $P(t)=P_0e^{kt}$, the time $T$ satisfying $2=e^{kT}$, so $T=\frac{\ln 2}{k}$.

Half-life

For exponential decay $P(t)=P_0e^{kt}$ ($k<0$), the time $T$ satisfying $\frac{1}{2}=e^{kT}$, so $T=\frac{\ln(1/2)}{k}$.

Newton’s Law of Cooling

A model $\frac{dT}{dt} = k(T-T_a)$ where temperature changes at a rate proportional to the difference from ambient temperature.

Ambient temperature (T_a)

The surrounding temperature in Newton’s Law of Cooling; the object’s temperature approaches this value over time.

Logistic differential equation

A population model $\frac{dP}{dt} = kP(1-\frac{P}{L})$ that grows quickly at first and levels off due to limited resources.

Carrying capacity (L)

The limiting population size in a logistic model; as P approaches L, growth slows toward zero.

Logistic equilibria

For $\frac{dP}{dt}=kP(1-\frac{P}{L})$, equilibria occur at $P=0$ and $P=L$ (where $\frac{dP}{dt}=0$).

Partial fractions (logistic solving step)

An algebra technique used after separation in logistic equations to integrate 1/[P(L−P)].

Logistic solution form

A standard explicit solution: P(t)= L/(1+Ce^{−kt}), with C determined by the initial condition.

Logistic maximum growth point

In logistic growth, dP/dt is greatest at P=L/2 (the inflection-point population).

Modeling template (rate statement)

Start by defining a changing quantity y(t), then write dy/dt = (expression for the rate), checking signs, units, and extreme behavior.