Unit 7: Differential Equations

Differential equation

An equation that relates an unknown function to one or more of its derivatives.

First-order differential equation

A differential equation whose highest derivative is the first derivative (e.g., dy/dt).

Rate form (of a differential equation)

Writing a DE as a rate, such as dy/dt = f(t, y), describing how a quantity changes.

Independent variable

The input variable (often x or t) with respect to which change is measured.

Dependent variable

The output variable (often y, P, or Q) whose value depends on the independent variable.

Solution (to a differential equation)

A function that makes the differential equation true for all relevant inputs.

General solution

A family of solutions containing an arbitrary constant (infinitely many functions), e.g., y = Ce^{3t}.

Constant of integration

The arbitrary constant (C) added after integrating, representing a family of antiderivatives/solutions.

Initial condition

A given value of the function at a specific input, such as y(0)=5.

Initial value problem (IVP)

A differential equation together with an initial condition used to determine a unique solution.

Particular solution

The single solution obtained after using an initial condition to determine the constant(s).

Implicit solution

A solution written as a relation between variables (not solved explicitly for y), e.g., ln|y| = 3t + C.

Checking (verifying) a proposed solution

Differentiate the candidate function and substitute into the DE to confirm both sides match.

Derivative notation (dy/dx and y′)

Common ways to write the derivative of y with respect to x: dy/dx or y′ (and dy/dt for t).

Slope field

A diagram of short line segments showing slopes f(x,y) at many points for a DE dy/dx = f(x,y).

Direction field

Another name for a slope field; it indicates the direction (slope) of solution curves at points.

Solution curve (on a slope field)

A curve that follows the slope field, staying tangent to the indicated segment slopes everywhere.

Tangent behavior in slope fields

A correct solution sketch must be tangent to the local slope segments rather than crossing them abruptly.

Isocline

A curve along which the slope value f(x,y) is constant (e.g., slope 0 where f(x,y)=0).

Autonomous differential equation

A DE where the rate depends only on the dependent variable, e.g., dy/dt = g(y).

Equilibrium solution

A constant solution y=c where dy/dx=0 whenever y=c (the solution stays at that value).

Stable equilibrium (attracting)

An equilibrium where nearby solutions move toward the equilibrium over time.

Unstable equilibrium (repelling)

An equilibrium where nearby solutions move away from the equilibrium over time.

Qualitative behavior

Describing solution behavior (increasing/decreasing, leveling off, approaching equilibria) without solving explicitly.

Euler’s method

A numerical method for approximating an IVP solution by stepping forward using tangent-line estimates.

Step size (h)

The x-increment used in Euler’s method; smaller h usually improves accuracy.

Euler update formulas

x{n+1}=xn+h and y{n+1}=yn + h f(xn, yn).

Left-endpoint slope (Euler)

Euler’s method uses the slope f(xn,yn) at the start (left endpoint) of each step.

Piecewise linear approximation

The graph formed by connecting Euler points with line segments, approximating the true solution curve.

Accumulated error (Euler)

Euler’s method is not exact; errors can build up step-by-step, especially with large h.

Separable differential equation

A DE that can be rearranged so all y-terms are on one side and all x- (or t-) terms on the other.

Separation of variables

Rewriting dy/dx=g(x)h(y) into (1/h(y))dy = g(x)dx so both sides can be integrated.

Antiderivative (in solving DEs)

An integral result used after separating variables to recover a function that matches the given rate of change.

SIPPY (separable/IVP checklist)

Separate, Integrate, Plus C, Plug in initial condition, Y equals (solve for y if needed).

Lost solution (when dividing by a variable)

Dividing by an expression like y can exclude the case y=0, so constant solutions must be checked.

Exponential growth/decay model

A model where the rate is proportional to the amount: dQ/dt = kQ.

Proportionality constant (k)

The constant in dQ/dt=kQ; k>0 gives growth and k<0 gives decay.

Exponential solution form

The solution to dQ/dt=kQ is Q(t)=Ce^{kt} (or Q(t)=Q0 e^{kt} with Q(0)=Q0).

Doubling time

For k>0 in Q(t)=Q0 e^{kt}, the time to double: T = (ln 2)/k.

Half-life

For k<0 in Q(t)=Q0 e^{kt}, the time to halve: T = ln(1/2)/k (a positive number).

Units of k

If t is in years, then k has units of “per year” so that kt is unitless in e^{kt}.

Logistic differential equation

A population model with limited resources: dP/dt = rP(1 - P/K).

Intrinsic growth rate (r)

In the logistic model, r>0 controls the growth speed when the population is small.

Carrying capacity (K)

In the logistic model, K>0 is the long-term limiting population level the solution approaches.

Logistic equilibria

Values where rP(1-P/K)=0, giving equilibrium solutions P=0 and P=K.

Maximum logistic growth at K/2

In dP/dt=rP(1-P/K), the growth rate is largest when P=K/2.

Partial fraction decomposition (logistic solving)

Splitting 1/[P(K-P)] into simpler fractions (like A/P + B/(K-P)) to integrate.

Logistic explicit solution form

An equivalent explicit logistic solution: P(t)=K/(1+Ae^{-rt}).

Finding A from an initial value

For P(t)=K/(1+Ae^{-rt}) with P(0)=P0, A=(K-P0)/P0.

Solving for time in a logistic model

To find t when P(t)=P1, isolate e^{-rt} first, then take ln and solve for t (avoids sign mistakes).