June 18, 2026 - Calculus 2 - Separable Differential Equations and Applications of Integration

Practice flashcards for review of differential equations, Newton's law of cooling, mass/density integrals, and surface area of revolution.

Equilibrium solution

A solution to a differential equation found by setting the function of $y$ equal to zero ($y' = 0$) and checking if it matches the initial condition.

Separable differential equation

A type of differential equation that can be solved by grouping the $y$ terms with $dy$ and the $x$ terms with $dx$, followed by integrating both sides.

Initial condition

A specific point, such as $(0, 3)$, used to solve for the constant $C$ to find a particular solution from a family of functions.

Newton's Law of Cooling

A principle stating that the rate of cooling of an object is proportional to the difference between its current temperature ($T$) and the temperature of its surroundings ($T_s$).

$$T_s$$

The variable in Newton's Law of Cooling representing the constant temperature of the surrounding environment, such as a kitchen.

Proportional constant ($k$)

A specific constant associated with an object's material properties that determines its rate of heat transfer in differential equations.

Family of functions

The set of all possible solutions to a differential equation, which occurs when a specific initial value has not yet been applied to determine the value of $C$.

Order of a differential equation

The value determined by the highest derivative present in the equation, not the power to which a derivative is raised.

Partial fraction decomposition

An integration technique used for rational functions by clearing the denominator and solving for constants like $A$ and $B$.

Linear density

The derivative of the mass function with respect to length, often denoted by the symbol $\rho$, where mass is defined as the integral of this function.

Radial density

A measure of density for circular objects that changes as a function of the radius ($x$) from the center.

Circumference for circular mass

The value $2\times\text{pi}\times x$ used in integrals to find the mass of a circular object by treating an inner circle as an unraveled wire of length $2\times\text{pi}\times x$.

Surface area of revolution

The area generated by revolving a curve around an axis, calculated using the formula $SA = \text{integral}(2\times\text{pi}\times f(x) \times \text{sqrt}(1 + (f'(x))^2) dx)$, where the square root captures the arc length component.

Exponential form

The method used to solve for $y$ in differential equations involving natural logs, such as rewriting $\text{ln}(y) = x + C$ as $y = e^{x+C}$.

Rational function

A function where the degree of the numerator and denominator determines the integration strategy, such as whether to use partial fractions or identifying natural log derivatives.