AP Calculus BC Flashcards: Modeling Situations With Differential Equations

What this deck covers

This deck focuses on Modeling Situations With Differential Equations, giving you a quick way to review the definitions, rules, and examples that matter most for AP Calculus BC.

How to use these flashcards

Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

Flashcard 1: Define a linear differential equation.

Answer: An equation where the dependent variable and derivatives appear linearly. Dependent variable appears to first power only.

Flashcard 2: Write the differential equation for exponential growth.

Answer: $\frac{dy}{dt} = ky$. Rate proportional to current amount.

Flashcard 3: What is the integrating factor for $dy/dx + 2y = e^x$?

Answer: $e^{2x}$. Exponential of integral of coefficient: $e^{\int 2 dx}$.

Flashcard 4: What is a particular solution?

Answer: A solution satisfying both the differential equation and initial conditions. Specific solution meeting initial conditions.

Flashcard 5: Identify the dependent variable in $dz/dt = z^2 + t^2$.

Answer: The dependent variable is $z$. Function being differentiated ($z$ depends on $t$).

Flashcard 6: Identify the dependent variable in $dy/dx = x^2 + y^2$.

Answer: The dependent variable is $y$. Variable being differentiated ($y$ depends on $x$).

Flashcard 7: What is the integrating factor for $y' + p(x)y = g(x)$?

Answer: $e^{\int p(x) \text{d}x}$. Multiplier that makes equation exact.

Flashcard 8: State a real-world example of a first-order differential equation.

Answer: Newton's Law of Cooling. Temperature change proportional to difference.

Flashcard 9: What is the general solution to $y'' + 4y = 0$?

Answer: $y = C_1 \text{cos}(2x) + C_2 \text{sin}(2x)$. Characteristic equation $r^2 + 4 = 0$ gives $r = ±2i$.

Flashcard 10: What method is used to solve $y' = ky$?

Answer: Separation of variables. Rearrange to separate $y$ and $x$ terms.

Flashcard 11: What is the general solution to $y'' + 9y = 0$?

Answer: $y = C_1 \text{cos}(3x) + C_2 \text{sin}(3x)$. Characteristic equation $r^2 + 9 = 0$ gives $r = ±3i$.

Flashcard 12: What does it mean for a function to be a solution to a differential equation?

Answer: Substituting it into the equation yields a true statement. The function satisfies the equation identically.

Flashcard 13: What is the characteristic equation of $y'' + 2y' + 5y = 0$?

Answer: $r^2 + 2r + 5 = 0$. Substitute $y'' = r^2$, $y' = r$ into equation.

Flashcard 14: Define a non-homogeneous differential equation.

Answer: An equation with non-zero terms not involving the dependent variable. Contains terms independent of dependent variable.

Flashcard 15: Identify the homogeneous part of $y'' + 5y' + 6y = e^x$.

Answer: $y'' + 5y' + 6y$. Left side when right side equals zero.

Flashcard 16: Solve $y' = \frac{1}{x}$ for $y$.

Answer: $y = \text{ln}|x| + C$. Antiderivative of $\frac{1}{x}$ is $\ln|x|$.

Flashcard 17: What function solves $y' - 2y = 0$?

Answer: $y = Ce^{2x}$. Exponential solution with coefficient $k = 2$.

Flashcard 18: What is the solution to $y'' - y = 0$?

Answer: $y = C_1e^x + C_2e^{-x}$. Characteristic equation $r^2 - 1 = 0$ gives $r = ±1$.

Flashcard 19: Identify the linear part of $y'' + 4y' + 4y = 5x$.

Answer: $y'' + 4y' + 4y$. Terms involving $y$ and its derivatives.

Flashcard 20: What is the solution for $dy/dx = y$ using separation of variables?

Answer: $y = Ce^{x}$. Separate variables: $\frac{dy}{y} = dx$, integrate.

Flashcard 21: What is the characteristic equation of $y'' + 4y' + 4y = 0$?

Answer: $r^2 + 4r + 4 = 0$. Replace $y''$ with $r^2$, $y'$ with $r$.

Flashcard 22: State the general solution form for $y' = ky$.

Answer: $y = Ce^{kt}$, where $C$ is a constant. Exponential growth/decay model solution.

Flashcard 23: Define a separable differential equation.

Answer: An equation where variables can be separated on opposite sides. Variables can be moved to opposite sides.

Flashcard 24: What is a homogeneous differential equation?

Answer: An equation where $0$ is a solution when all terms set to $0$. All terms involve the dependent variable.

Flashcard 25: Identify the type of differential equation: $y'' + y = \tan(x)$.

Answer: Second-order non-homogeneous differential equation. Second-order with non-zero right side.

Flashcard 26: Solve for $y$ in the differential equation $\frac{dy}{dx} = 0$.

Answer: $y = C$, where $C$ is a constant. Zero derivative means constant function.

Flashcard 27: What is the order of $d^2y/dx^2 + 3y' + 4y = x^2$?

Answer: Order is 2. Highest derivative is second order.

Flashcard 28: Identify the independent variable in $dy/dx = x^2 + y^2$.

Answer: The independent variable is $x$. Variable with respect to which we differentiate.

Flashcard 29: What type of differential equation is $y' + p(x)y = g(x)$?

Answer: First-order linear differential equation. Standard form with $y$ and $y'$ terms only.

Flashcard 30: Find the derivative of $y = Ce^{kt}$.

Answer: $y' = kCe^{kt}$. Apply chain rule: $\frac{d}{dt}(Ce^{kt}) = kCe^{kt}$.