AP Calculus AB 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 AB.

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 solution to $y' = y$ with $y(0) = 2$?

Answer: $y = 2e^x$. Exponential solution $y = Ce^x$ with $C = 2$ from condition.

Flashcard 2: What is a solution to $y' = y$ with $y(0) = 2$?

Answer: $y = 2e^x$. Exponential solution $y = Ce^x$ with $C = 2$ from condition.

Flashcard 3: Find the particular solution of $y' = 2x$ given $y(1) = 3$.

Answer: $y = x^2 + 2$. Integrate $2x$ to get $x^2 + C$, then use initial condition.

Flashcard 4: Identify the particular solution of $y' = 2y$ given $y(1) = 3$.

Answer: $y = 3e^{2x - 2}$. Apply exponential solution and initial condition at $x = 1$.

Flashcard 5: What is the solution to $y' = 3y$ with $y(0) = 5$?

Answer: $y = 5e^{3x}$. Exponential solution with growth constant 3 and initial value 5.

Flashcard 6: What is the solution to $dy/dx = 3x^2$ with $y(0) = 1$?

Answer: $y = x^3 + 1$. Integrate and apply the given initial condition.

Flashcard 7: What does it mean for a DE to be linear?

Answer: Dependent variable and its derivatives are linear. No powers or products of $y$ or its derivatives.

Flashcard 8: Identify the independent variable in $dy/dt = -2y$.

Answer: $t$. The variable in the denominator of the derivative.

Flashcard 9: State the general solution for $y' = \frac{1}{2}y$.

Answer: $y = Ce^{\frac{1}{2}x}$. Exponential solution with growth rate $k = \frac{1}{2}$.

Flashcard 10: Identify the order of the differential equation $y'' + 3y' - 5y = 0$.

Answer: Second order. The highest derivative is $y''$ (second derivative).

Flashcard 11: Identify the dependent variable in $dy/dx = x^2y$.

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

Flashcard 12: What is the integrating factor for $y' + py = q$?

Answer: $e^{\int p \, dx}$. Multiplying factor that makes the left side an exact derivative.

Flashcard 13: Identify the dependent variable in $dy/dx = x^2y$.

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

Flashcard 14: What method is used to solve $y' = xy$?

Answer: Separation of variables. Can separate variables: $\frac{dy}{y} = x dx$.

Flashcard 15: What is a particular solution of a differential equation?

Answer: A solution satisfying the initial conditions. General solution with constants determined by conditions.

Flashcard 16: What is the solution to $\frac{dy}{dx} = 3x^2$ with $y(0)=4$?

Answer: $y = x^3 + 4$. Integrate $3x^2$ and apply the initial condition.

Flashcard 17: What is an initial condition in differential equations?

Answer: A value that specifies the solution of a DE at a point. Determines a unique solution from the general solution.

Flashcard 18: What does it mean for a DE to be separable?

Answer: It can be written as $g(y)dy = f(x)dx$. Variables can be moved to opposite sides for integration.

Flashcard 19: Identify the independent variable in $dy/dt = -2y$.

Answer: $t$. The variable in the denominator of the derivative.

Flashcard 20: What is the order of the DE $x^2y'' + xy' + y = 0$?

Answer: Second order. Highest derivative present is the second derivative $y''$.

Flashcard 21: What is the integrating factor for $y' + py = q$?

Answer: $e^{\int p \, dx}$. Multiplying factor that makes the left side an exact derivative.

Flashcard 22: Identify the particular solution of $y' = 2y$ given $y(1) = 3$.

Answer: $y = 3e^{2x - 2}$. Apply exponential solution and initial condition at $x = 1$.

Flashcard 23: What is the general solution of $y' = ky$?

Answer: $y = Ce^{kx}$. Exponential growth/decay solution form.

Flashcard 24: What is the integrating factor for $y' + 3y = 0$?

Answer: $e^{3x}$. For homogeneous equation, integrating factor is $e^{3x}$.

Flashcard 25: What is a homogeneous differential equation?

Answer: All terms are a function of the dependent variable and its derivatives. No external forcing term, only function and derivatives.

Flashcard 26: What is the order of the DE $x^2y'' + xy' + y = 0$?

Answer: Second order. Highest derivative present is the second derivative $y''$.

Flashcard 27: Identify the order of the differential equation $y'' + 3y' - 5y = 0$.

Answer: Second order. The highest derivative is $y''$ (second derivative).

Flashcard 28: Identify the solution to $y' = 4y$ with $y(0) = 1$.

Answer: $y = e^{4x}$. Exponential solution $y = Ce^{4x}$ with $C = 1$.

Flashcard 29: What technique solves $y' + p(x)y = q(x)$?

Answer: Integrating factor. Standard method for linear first-order equations.

Flashcard 30: Which method is used to solve $y' = yx$?

Answer: Separation of variables. Variables can be separated: $\frac{dy}{y} = x dx$.

Flashcard 31: What method is used to solve $y' = xy$?

Answer: Separation of variables. Can separate variables: $\frac{dy}{y} = x dx$.

Flashcard 32: What is an initial condition in differential equations?

Answer: A value that specifies the solution of a DE at a point. Determines a unique solution from the general solution.

Flashcard 33: Find the particular solution of $y' = 2x$ given $y(1) = 3$.

Answer: $y = x^2 + 2$. Integrate $2x$ to get $x^2 + C$, then use initial condition.

Flashcard 34: What is the form of a linear first-order DE?

Answer: $y' + p(x)y = q(x)$. Standard form for first-order linear differential equations.

Flashcard 35: What does it mean for a DE to be linear?

Answer: Dependent variable and its derivatives are linear. No powers or products of $y$ or its derivatives.

Flashcard 36: What is the general solution of $y' = 0$?

Answer: Constant function. No change means the function remains constant.

Flashcard 37: What is the solution to $y' + y = 0$ with $y(0) = 3$?

Answer: $y = 3e^{-x}$. Exponential decay with initial condition applied.

Flashcard 38: Identify the solution to $y' = 4y$ with $y(0) = 1$.

Answer: $y = e^{4x}$. Exponential solution $y = Ce^{4x}$ with $C = 1$.

Flashcard 39: What is a first-order differential equation?

Answer: An equation involving the first derivative of a function. Only involves $y'$, no higher derivatives.

Flashcard 40: What is a particular solution for $y' = xy$ given $y(1) = 4$?

Answer: $y = 4e^{\frac{x^2 - 1}{2}}$. Separate variables and integrate, then apply condition.

Flashcard 41: What is a particular solution of a differential equation?

Answer: A solution satisfying the initial conditions. General solution with constants determined by conditions.

Flashcard 42: What is a particular solution for $y' = xy$ given $y(1) = 4$?

Answer: $y = 4e^{\frac{x^2 - 1}{2}}$. Separate variables and integrate, then apply condition.

Flashcard 43: Which function satisfies $y' = 0$?

Answer: Constant functions. Zero derivative means no change, so $y$ is constant.

Flashcard 44: What is a differential equation?

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

Flashcard 45: What is the solution to $y' + y = 0$ with $y(0) = 3$?

Answer: $y = 3e^{-x}$. Exponential decay with initial condition applied.

Flashcard 46: What is the general solution of $y' = 0$?

Answer: Constant function. No change means the function remains constant.

Flashcard 47: What is the solution to $dy/dx = 3x^2$ with $y(0) = 1$?

Answer: $y = x^3 + 1$. Integrate and apply the given initial condition.

Flashcard 48: Which function satisfies $y' = 0$?

Answer: Constant functions. Zero derivative means no change, so $y$ is constant.

Flashcard 49: What is a first-order differential equation?

Answer: An equation involving the first derivative of a function. Only involves $y'$, no higher derivatives.

Flashcard 50: What is the solution to $y' = 3y$ with $y(0) = 5$?

Answer: $y = 5e^{3x}$. Exponential solution with growth constant 3 and initial value 5.

Flashcard 51: State the form of a separable differential equation.

Answer: $\frac{dy}{dx} = g(x)h(y)$. Variables can be separated to each side of the equation.

Flashcard 52: What technique solves $y' + p(x)y = q(x)$?

Answer: Integrating factor. Standard method for linear first-order equations.

Flashcard 53: What is the solution to $\frac{dy}{dx} = 3x^2$ with $y(0)=4$?

Answer: $y = x^3 + 4$. Integrate $3x^2$ and apply the initial condition.

Flashcard 54: What is a homogeneous differential equation?

Answer: All terms are a function of the dependent variable and its derivatives. No external forcing term, only function and derivatives.

Flashcard 55: What is the form of a linear first-order DE?

Answer: $y' + p(x)y = q(x)$. Standard form for first-order linear differential equations.

Flashcard 56: What is a differential equation?

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

Flashcard 57: What does it mean for a DE to be separable?

Answer: It can be written as $g(y)dy = f(x)dx$. Variables can be moved to opposite sides for integration.

Flashcard 58: Find the integrating factor for $y' + 2y = 3x$.

Answer: $e^{2x}$. For $y' + py = q$, integrating factor is $e^{\int p dx} = e^{2x}$.

Flashcard 59: What is the integrating factor for $y' + 3y = 0$?

Answer: $e^{3x}$. For homogeneous equation, integrating factor is $e^{3x}$.

Flashcard 60: What is the general solution of $y' = ky$?

Answer: $y = Ce^{kx}$. Exponential growth/decay solution form.

Flashcard 61: Which method is used to solve $y' = yx$?

Answer: Separation of variables. Variables can be separated: $\frac{dy}{y} = x dx$.

Flashcard 62: State the form of a separable differential equation.

Answer: $\frac{dy}{dx} = g(x)h(y)$. Variables can be separated to each side of the equation.

Flashcard 63: Find the integrating factor for $y' + 2y = 3x$.

Answer: $e^{2x}$. For $y' + py = q$, integrating factor is $e^{\int p dx} = e^{2x}$.

Flashcard 64: State the general solution for $y' = \frac{1}{2}y$.

Answer: $y = Ce^{\frac{1}{2}x}$. Exponential solution with growth rate $k = \frac{1}{2}$.