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: What is a differential equation?

Answer: An equation involving derivatives of a function. Relates a function to its rate of change.

Flashcard 2: 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 3: 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 4: What is the integrating factor for $dy/dx + 2y = e^x$?

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

Flashcard 5: 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 6: Identify the independent variable in $dz/dt = z^2 + t^2$.

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

Flashcard 7: What is a differential equation?

Answer: An equation involving derivatives of a function. Relates a function to its rate of change.

Flashcard 8: Solve $y' = \frac{1}{x}$ for $y$.

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

Flashcard 9: What is the integrating factor for $y' + p(x)y = g(x)$?

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

Flashcard 10: 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 11: Solve the differential equation $\frac{dy}{dx} = 3x^2$.

Answer: $y = x^3 + C$. Direct integration: $\int 3x^2 dx = x^3 + C$.

Flashcard 12: Identify the homogeneous part of $y'' + 5y' + 6y = e^x$.

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

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: 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 15: 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 16: What is the solution to $y'' - y = 0$?

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

Flashcard 17: What is the solution for $dy/dx = y$ using separation of variables?

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

Flashcard 18: What is the order of $d^2y/dx^2 + 3y' + 4y = x^2$?

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

Flashcard 19: What method is used to solve $y' = ky$?

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

Flashcard 20: What function solves $y' - 2y = 0$?

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

Flashcard 21: 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 22: Write the differential equation for exponential growth.

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

Flashcard 23: Solve the differential equation $\frac{dy}{dx} = 3x^2$.

Answer: $y = x^3 + C$. Direct integration: $\int 3x^2 dx = x^3 + C$.

Flashcard 24: What is the solution for $dy/dx = y$ using separation of variables?

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

Flashcard 25: 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 26: 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 27: 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 28: Define a separable differential equation.

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

Flashcard 29: 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 30: 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 31: 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 32: Solve for $y$ in $dy/dx = 2xy$ using separation of variables.

Answer: $y = Ce^{x^2}$. Separate: $\frac{dy}{y} = 2x dx$, integrate both sides.

Flashcard 33: What function solves $y' - 2y = 0$?

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

Flashcard 34: 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 35: 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 36: State the general solution form for $y' = ky$.

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

Flashcard 37: Identify the linear part of $y'' + 4y' + 4y = 5x$.

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

Flashcard 38: 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 39: Identify the linear part of $y'' + 4y' + 4y = 5x$.

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

Flashcard 40: State a real-world example of a first-order differential equation.

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

Flashcard 41: Identify the order of the differential equation $y'' + 3y' - 4y = 0$.

Answer: Order is 2. Highest derivative is second order ($y''$).

Flashcard 42: What is the order of the equation $d^3y/dx^3 + y = 0$?

Answer: Order is 3. Highest derivative is third order.

Flashcard 43: 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 44: 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 45: 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 46: 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 47: Solve for $y$ in $dy/dx = 2xy$ using separation of variables.

Answer: $y = Ce^{x^2}$. Separate: $\frac{dy}{y} = 2x dx$, integrate both sides.

Flashcard 48: 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 49: What is the integrating factor for $dy/dx + 2y = e^x$?

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

Flashcard 50: 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 51: 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 52: Solve $y' = \frac{1}{x}$ for $y$.

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

Flashcard 53: 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 54: State the general solution form for $y' = ky$.

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

Flashcard 55: Identify the independent variable in $dz/dt = z^2 + t^2$.

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

Flashcard 56: 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 57: Find the derivative of $y = Ce^{kt}$.

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

Flashcard 58: State a real-world example of a first-order differential equation.

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

Flashcard 59: What is the order of the equation $d^3y/dx^3 + y = 0$?

Answer: Order is 3. Highest derivative is third order.

Flashcard 60: What is a particular solution?

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

Flashcard 61: 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 62: 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 63: Find the derivative of $y = Ce^{kt}$.

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

Flashcard 64: What method is used to solve $y' = ky$?

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

Flashcard 65: Define a linear differential equation.

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

Flashcard 66: Write the differential equation for exponential growth.

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

Flashcard 67: Define a separable differential equation.

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

Flashcard 68: What is the order of $d^2y/dx^2 + 3y' + 4y = x^2$?

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

Flashcard 69: Identify the order of the differential equation $y'' + 3y' - 4y = 0$.

Answer: Order is 2. Highest derivative is second order ($y''$).

Flashcard 70: Define a linear differential equation.

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