Separable Differential Equations and Applications Flashcards

Vocabulary practice for first-order separable differential equations and their application in brine tank mixture problems.

Separable Differential Equation

A first order differential equation that can be written in the form $y' = f(x) \times g(y)$, allowing variables to be separated on opposite sides of the equation.

First Order Differential Equation

A differential equation that contains exactly one derivative, such as $y'$ or $\frac{dy}{dx}$.

Equilibrium Solutions

Functions where $y$ is equal to a constant and the derivative is zero, appearing as constant horizontal lines in a direction field.

Differentials

The terms $dy$ and $dx$ which mathematically refer to infinitesimal changes in the variables $y$ and $x$, respectively.

Partial Fraction

A method used to integrate rational functions, such as breaking down the denominator $y^2 - 4$ into $(y - 2)$ and $(y + 2)$ to solve for constants $A$ and $B$.

General Solution

A solution to a differential equation that includes arbitrary constant terms, such as $+C$, representing an infinite number of possible functions.

Particular Solution

A specific solution to a differential equation where the constants (like $C$) have been determined by applying an initial condition.

Initial Condition

A specified value for the function at a specific time or point, such as $y(0) = 3$, used to solve for unique constants in a differential equation.

Extraneous Solution

A value found during calculation that cannot be a valid solution, such as a zero in a numerator that also causes the denominator of a rational function to be zero.

Inflow Rate

In concentration problems, the amount of salt entering a tank, calculated by multiplying the solution's flow rate (e.g., $2 \text{ L/min}$) by its salt concentration (e.g., $0.5 \text{ kg/L}$).

Outflow Rate

The rate at which salt leaves a tank, calculated by multiplying the rate of liquid exiting the tank by the current concentration of salt in the mixture.

Concentration

The amount of salt currently in the solution divided by the total volume of the solution, often expressed as $\frac{S(t)}{100}$ in tank problems.

Natural Logarithm ($\text{ln}$)

The function resulting from the integration of $\frac{1}{y}$, which is typically removed by raising both sides of an equation as powers of the base $e$.