Unit 7: Mastery of Differential Equations

Separation of Variables

An algebraic technique used to solve first-order differential equations by separating variables.

General Solution

A family of functions containing an arbitrary constant of integration ($+C$).

Particular Solution

A specific function obtained by solving for the constant of integration using an initial condition.

Initial Condition

A specific condition that helps determine the value of the arbitrary constant in a solution.

Integrate

To find the antiderivative of a function.

Exponential Growth

When the rate of change of a quantity is directly proportional to the quantity itself, resulting in increase.

Exponential Decay

When the rate of change of a quantity is directly proportional to the quantity itself, resulting in decrease.

Logistic Growth

A model of population growth that accounts for limited resources and carrying capacity.

Carrying Capacity

The maximum population size that an environment can sustain.

Differential Equation

An equation that relates a function with its derivatives.

Partial Fraction Decomposition

A method used to break down a complex fraction into simpler parts to facilitate integration.

Antiderivative

A function whose derivative is the given function.

$y(t)$

The quantity at time $t$ in the context of differential equations.

$k$

The constant of proportionality in exponential models.

Adding Constant

The step of adding a constant after integrating a function.

$M$ in Logistic Growth

The carrying capacity in a logistic growth model.

Inflection Point

The point where the function changes concavity, indicating the maximum growth rate in logistics.

Misplacement of Constant Error

An error of adding constant $C$ after algebra rather than immediately after integrating.

Absolute Value in Integrals

The requirement to use absolute values when integrating functions that can take negative values.

Proportional Growth Rate

A growth rate that is a constant fraction of the current value.

Standard Form for Exponential Models

The form: $\frac{dy}{dt} = ky$ for exponential growth or decay.

Separation Technique Steps

1. Separate variables; 2. Integrate each side; 3. Add constant; 4. Use initial condition; 5. Isolate $y$.

Logistic Equation Form 1

$\frac{dP}{dt} = kP(1 - \frac{P}{M})$.

Logistic Equation Form 2

$\frac{dP}{dt} = cP(M - P)$.

Maximum Growth Rate Condition

Occurs when population $P$ is half the carrying capacity.

Initial Value

The amount or quantity at the starting point in a model.

Decay Rate

The rate at which a substance decreases over time due to decay.